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DG structures on categorified quantum groups

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DG structures on categorified quantum groups

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Content DG STRUCTURES ON CATEGORIFIED QUANTUM GROUPS A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) Ilknur Sever Egilmez August 2018 c Copyright by Ilknur Sever Egilmez 2018 All Rights Reserved ii To my beloved Hilmi iii Acknowledgments I would like to express my sincere gratitude towards my advisor, Professor Aaron Lauda, for his constant guidance, support and understanding during the course of my studies at the University of Southern California (USC). It has been a pleasure and privilege to work with Professor Lauda who helped me grow as an independent researcher. This thesis would not be possible without his help and support. Also, I would like to thank Professor Jason Fulman and Professor Antonio Ortega for serving on my dissertation committee. Moreover, I would like to express my thanks to David Rose, Matthew Hogancamp, Andrea Appel for mentoring me and collaborating with me on the projects, and You Qi for his helpful discussions. I thank all my friends in the Mathematics department especially, Can Ozan Oguz, Ezgi Kantarci, Enes Ozel, Ozlem Ejder, Melike Sirlanci, Guher Camliyurt and my friends Umit Bas, Hacer Sifanur, Onur Tuysuzoglu, Vanessa Landes, Ahsan Javed for their friendship and support which made the PhD life more enjoyable. I thank to my family and especially to my beloved husband Hilmi whose love, encouragement, support, and understanding motivated me throughout my PhD studies. iv Contents Acknowledgments iv 1 Introduction 1 1.1 Categorication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Quantum Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Odd Dierentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.1 Motivations from link homology theory . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 The oddication program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3 Covering Kac-Moody algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.4 Dierential graded structures on categoried quantum group . . . . . . . . . 12 1.3.5 Categorication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 A DG-extension of Symmetric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 A DG-Structure on Odd 2-category 17 2.1 Super dg theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.1 Super 2-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.2 Super dg-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.3 Super dg-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.1.4 Super dg-2-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Hopfological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.1 Basic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.2 DG-algebras from the Hopfological perspective . . . . . . . . . . . . . . . . . 24 v 2.2.3 Decategorication from the Hopfological perspective . . . . . . . . . . . . . . 25 2.2.4 Gaussian integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Results on Grothendieck groups of super dg-algebras . . . . . . . . . . . . . . . . . . 28 2.3.1 Grothendieck group of a super dg-algebra . . . . . . . . . . . . . . . . . . . . 28 2.3.2 Positively graded dg-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.3 Grothendieck ring of super dg-2-categories . . . . . . . . . . . . . . . . . . . 29 2.3.4 Fantastic ltrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4 The odd 2-category for sl(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4.1 The odd nilHecke ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4.2 The odd categoried quantum group . . . . . . . . . . . . . . . . . . . . . . . 34 2.4.3 Additional Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.5 Derivations on the odd 2-category . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.5.1 Derivations on odd 2-category arising from the fantastic ltration . . . . . . . 57 2.6 Dierentials on the odd 2-category . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.7 Quantum sl(2) at a fourth root of unity . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.7.1 Idempotented quantum sl 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.7.2 Small quantum sl 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.8 Categorication results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.8.1 Divided power modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.8.2 The DG-Grothendieck ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3 A DG-extension of Symmetric Functions 76 3.1 The nilHecke algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.1.1 The denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.2 The extended nilHecke algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.2.1 The denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.2.2 Action on polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.2.3 Dierentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.3 The ring of extended symmetric polynomials . . . . . . . . . . . . . . . . . . . . . . 83 3.3.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 vi 3.3.2 The size of extended symmetric functions . . . . . . . . . . . . . . . . . . . . 85 3.3.3 Structure of the extended nilHecke algebra . . . . . . . . . . . . . . . . . . . 93 3.3.4 Bases of ext n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.3.5 Combinatorial identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.4 Solomon's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.4.1 Superpolynomials and superinvariants . . . . . . . . . . . . . . . . . . . . . . 106 3.4.2 Action of the extended nilHecke algebra . . . . . . . . . . . . . . . . . . . . . 107 3.4.3 Preliminary computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.4.4 NH ext n {equivariant isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.4.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.5 Dierentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.5.1 The standard dierential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.5.2 Koszul complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.5.3 Deformed dierentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.5.4 Categorication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4 Conclusions and Future Work 129 Bibliography 131 vii List of Tables 2.1 Comparison between DG-algebras and Hophological algebra . . . . . . . . . . . . . . 24 3.1 Example for n = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 viii List of Figures 1.1 Illustration of categorication and decategorication . . . . . . . . . . . . . . . . . . 2 1.2 Illustration of general form of categorication and decategorication . . . . . . . . . 3 1.3 Action of E and F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Decomposition of an n-bubble for taking derivation . . . . . . . . . . . . . . . . . . . 49 ix Abstract This thesis studies DG structures on categoried quantum groups. In the rst part of the thesis, we equip Ellis and Brundan's version of the odd categoried quantum group forsl 2 with a dierential and identify the Grothendieck ring of the associated dg-2-category with a categorication of sl 2 at a fourth root of unity. In the second part of the thesis we investigate analogs of symmetric functions arising from an extension of the nilHecke algebra dened by Naisse and Vaz. These extended symmetric functions form a subalgebra of the polynomial ring tensored with an exterior algebra. We dene families of bases for this algebra and show that it admits a family of dierentials making it a sub- DG-algebra of the extended nilHecke algebra. The ring of extended symmetric functions equipped with this dierential is quasi-isomorphic to the cohomology of a Grassmannian. We also introduce new deformed dierentials on the extended nilHecke algebra that when restricted makes extended symmetric functions quasi-isomorphic to GL(N)-equivariant cohomology of Grassmannians. Chapter 1 Introduction In this thesis, we study dierentials on categoried quantum groups for the simple lie algebra sl 2 . After we equip the categoried quantum group with a dierential graded structure, we decategorify it by taking the Grothendieck ring. The thesis consists of two main parts. The rst part focuses on the study of dierentials on the odd 2-category U for the simple lie algebra sl 2 . We classify derivations on this odd 2-category U, which is dened by Brundan and Ellis in [BE17b], by extending the dierential on the odd nilHecke algebra given in [EQ16c]. Moreover, we classify the derivations which give rise to a fantastic ltration, which is an essential tool in our work to determine the Grothendieck ring of DG-category. Then, we show that the resulting derivation is a dierential which gives rise to a fantastic ltration. In the last part, we conjecture that odd DG 2-category (U;@) categorifes the small quantum group at forth root of unity. The detailed version of this part is given in [EL18] with the proof of this conjecture. In the second part of the thesis, we equip the ring of extended symmetric polynomials with a family of dierentials d N , such that with this DG structure, the ring of extended symmetric polynomials forms a sub-DG-algebra of extended nilHecke algebra which arise in categorication of Verma modules in [NV16]. We also show that the ring of extended nilHecke algebra can be equipped with a family of deformed dierentials which also restricts to the ring of extended symmetric polynomials and the resulting cohomology of the ring of extended symmetric polynomials with deformed dierentials is isomorphic to the equivariant cohomology of Grassmannian. Furthermore, 1 CHAPTER 1. INTRODUCTION 2 we study some combinatorial properties of the family of bases, which we dene for the ring of extended symmetric polynomials. The outline of the rest of the thesis is as follows. In the remaining sections of the rst chapter we give some preliminary background on categorication and quantum groups and their categorication, as it is the main object we work with in this thesis. In Section 1.3 and Section 1.4, we summarize main contributions of this thesis. In the second chapter, we study the odd DG-2-category (U;@) in detail. The third chapter studies the the ring of extended symmetric polynomials and family of bases we dened on them with some combinatorical results, family of dierentials on the ring of extended symmetric polynomials and the ring of extended nilHecke algebra arising from the study of categorication of Verma modules in [NV16]. The last chapter presents concluding remarks and future work. We believe that the results we presented in this thesis could be generalized for lie algebras of dierent types, and also one can study decategorication using Trace which is dened in Section 1.1. 1.1 Categorication Categorication can be thought of as a tool turning sets into categories. Set has one-layer infor- mation, namely elements, however a category has two-layer information, objects and morphisms between objects. Categorication takes this one-layer object and turns it into a two-layer object by adding another layer of information and enhancing its structure. Decategorication, which is an inverse of the categorication process, forgets this added extra structure and simplies category to set. Figure 1.1 below illustrates this procedure. set category "" Categorication bb Decategorication Figure 1.1: Illustration of categorication and decategorication In general, categorication is a functor from n-category to (n + 1)-category, which takes the CHAPTER 1. INTRODUCTION 3 algebraic structure of the n-category and by adds another layer of information by raising it to a more enriched and complex version (n + 1)-category. In other words, n-category can be viewed as the shadow of the (n + 1)-category, and as a shadow it lacks some of the properties more specically complexity and richness of (n+1)-category. Categorication procedure should be considered together with decategorication, which is a map designed to forget the some of the structure of the (n + 1)-category and reduce it to an n-category. Figure 1.2 below illustrates the categorication and decategorication procedure in general. 0-category 1-category "" Categ. bb Decat. 2-category "" bb Categ. Decat. OO Objects OO Objects 1-morphisms OO Objects 1-morphisms 2-morphisms Figure 1.2: Illustration of general form of categorication and decategorication Note that set in Figure 1.1 corresponds to 0-category in above Figure 1.2 which has elements as 0-objects, and by a category we mean 1-category. One of the best examples to illustrate this notion is the categorication of natural numbers N via Vect k , the category of nite dimensional vector spaces over a eld k. In this example N is 0- category and Vect k is 1-cateogry, where objects are nite dimensional vector spaces and morphisms are linear maps. Any natural number n2 N is categoried by a vector space V of dimension n, and decategorication of any nite dimensional vector space V is taking the dimension dim(V ). Algebraic structure of N is lifted to Vect k with categorication in the following way. Sum of two numbersn +m is categoried to a direct sumV n V m of two vector spacesV n andV m of dimension n andm respectively, so when you take the dimension you get dim(V n V m ) =n +m, and product of two numbersnm is categorifed by tensor productV n V m of vector spacesV n andV m , and when one takes the dimension dim(V n V m ) =nm, then recovers the product of numbers nm. Category Vect k of nite dimensional vector spaces has more complex structure than the the set of natural CHAPTER 1. INTRODUCTION 4 numbers N, in addition to carrying the underlying algebraic properties of N as described above, it also has the space of linear maps between any given two vector spaces V and W . For more details on the categorication, we refer the reader to [Lau12, BD98, KMS07, Sav14]. An important example of the categorication procedure, which also motivates the rst part of this thesis, is the categorication of knot invariants. Knots can be dened as embedding of S 1 into R 3 or into S 3 , the three dimensional sphere. The main question in knot theory, which is still an important open problem, is to understand if given any two knot diagrams represent the same knot or not. There are many invariants dened for knots or links in order to solve this problem. Some of examples of these invariants are polynomial link invariants such as Jones polynomial, Alexander polynomial, Conway polynomial, HOMFLYPT polynomial, and link homology theories including Khovanov homology, knot Floer homology. For an introductory survey on knot theory, we refer to [PS97]. Jones polynomialsJ(K), which is dened for a knot K or a link, is a Laurent polynomial Z[q;q 1 ] and one of the polynomial knot invariants as mentioned above. However, it is not a complete knot invariant, which means if for any two knots K 1 and K 2 , we have J(K 1 ) = J(K 2 ) then we can not conclude thatK 1 is isotopic toK 2 . A categorication of Jones polynomial was given by Khovanov in [Kho00] was the rst step in categoried link invariants, where he associated a graded link homology theoryKh(K) to a knot (or a link)K whose graded Euler characteristic is the Jones polynomial. A homoloygy theory associated to a knot is more powerful than a polynomial as an invariant, in this case Khovanov homology is functorial as opposed to Jones polynomial. Jones polynomial is an example of a quantum knot invariant, which is dened using representa- tion theory of quantum groupU q (sl 2 ). Knot invariants which arise from the study of representation theory of quantum groupsU(g) associated to Lie algebra g are called Rashetikhin-Turaev invariants, see [RT90]. With choosing dierent Lie algebras or studying various dimensional representations of those lie algebras give dierent polynomial knot invariants. For example, if we choose the Lie algebra sl 2 , then its dening two dimensional representation is Jones polynomial, (n + 1) dimen- sional representation is colored Jones polynomial. If we consider the Lie algebra sl n then we get HOMFLYPT polynomial. Categorication and decategorication are not uniquely dened. A categoryC can have more CHAPTER 1. INTRODUCTION 5 than one categorication, and similarly there can be more than one way to decategory it. The most commonly used decategorication functor is taking the Groethendieck group K 0 . For any additive categoryC, its split Groethendieck group K 0 (C) is a free abelian group generated by isomorphism classes of objects modulo the relation [AB] = [A] + [B] K 0 (C) =f[X]jX2 Ob (C)g= There is another decategorication functor called Trace decategorication, which instead of tak- ing the isomorphism classes of objects takes the isomorphism classes of morphisms. For any linear categoryC, Trace Tr(C) is an abelian group dened as Tr(C) = M x2Ob(C) C(x;x)=Spanffggfg where f : x! y and g : y! x, for any x;y2C. For more on the trace decategorication and comparison between Tr and K 0 , see [BHLZ14, BGHL, Ziv14]. In this thesis, decategorication is taking Grothendieck ring for us, however we think that trace decategorication of a dierential graded category, i.e. DG-category, and in our case of an odd DG-2-category is an interesting problem. As trace of a category, by denition, has a potential to have richer structure. We propose it as a future work in Chapter 4. 1.2 Quantum Groups A quantum group is a Hopf algebra given by generators and relations. For a lie algebra g, its quantum group, denoted as U q (g), is dened by one parameter deformation of the universal enveloping algebra U(g) of the lie algebra g. For example, the simple lie algebra sl 2 , which is the algebra of two-by-two traceless matrices over a led k, generated byfE;F;Hg where, E = 2 6 4 0 1 0 0 3 7 5 ; F = 2 6 4 0 0 1 0 3 7 5 ; H = 2 6 4 1 0 0 1 3 7 5 CHAPTER 1. INTRODUCTION 6 with the commutation relations [E;F ] =H; ; [H;E] = 2E ; [H;F ] =2F and its universal enveloping algebra U(sl 2 ) is the associative algebra with the same generators as the simple lie algebra sl 2 with relations EFFE =H ; HEEH = 2E ; HFFH =2F where the lie bracket [a;b] is replaced byabba for anya;b2 sl 2 . Then its one parameter deformed quantum group U q (sl 2 ) is obtained by replacing the generator H byK andK 1 with the relations KK 1 =K 1 K = 1 KE =q 2 EK KF =q 2 FK EFFE = KK 1 qq 1 and the quantum group U q (sl 2 ) is algebra over the ring of rational functionsQ(q) with generators E;F;K;K 1 and given relations above. For any U q (sl 2 )-moduleV and any scalar6= 0, we denoteV the subspace of all vectorsv2V such thatKv =q v. We call the scalar weight ofV ifV 6= 0. A nite dimensional representation V of U q (sl 2 ) can be decomposed into eigenspaces v by the action of K. It can be shown that weights 2 Z by using the representation of sl 2 . We consider the representations of U q (sl 2 ) which admits a weight space decomposition. By using the dening relation of U q (sl 2 ), we can dene the actions of E and F as follows, for any v2V , K(Ev) =q 2 E(Kv) =q +2 Ev K(Fv) =q 2 F (Kv) =q 2 Fv which implies E :V !V +2 and F :V !V 2 . Figure 1.3 below illustrates the actions of E and CHAPTER 1. INTRODUCTION 7 F . For any V = L n2Z V n nite dimensional representation of U q (sl 2 ), we have V n2 V n V n+2 E E __ __ F F Figure 1.3: Action of E and F The main sl 2 relation becomes (EFFE)v = KK 1 qq 1 v = q q v = []v where [] is called the quantum integer . Since we are interested in the representations of U q (sl 2 ) which admit a weight decomposition, there is a modied version _ U of U q (sl 2 ) and it can be found in Section 2.7.1. For more on the quantum groups and their representation see, [Kas12, CP94, Lus10]. Existence of categorical actions of _ U, suggested the categorication of U q (sl 2 ) should exist. In a categorical _ U-action, one replaces the vector spaces V by additive categoriesV , which are triangulated so that they are equipped with an auto-equivalencef1g :V !V , and linear maps are replaced by functors 1 :V !V ,E1 :V !V +2 , andF1 :V !V 2 . The main sl 2 relation in U q (sl 2 ) is lifted to a natural isomorphisms of functors. EF1 =FE1 1 [] ; for n 0 FE1 =EF1 1 [] ; for n 0 where 1 [] := 1 1 1 3 1 1 Weight categoriesV categorifyQ(q)-vector spaces V of some representation V = L 2Z V of _ U with decategorication K 0 (V ) Z[q;q 1 ] Q(q) =V and the additive functors E1 andF1 onV induce the action of E1 and F1 on the split CHAPTER 1. INTRODUCTION 8 Grothendieck group [E1 ]; [F1 ] :K 0 (V)!K 0 (V): Existence of these categorical actions suggested the categorifcation of quantum groups should ex- ist, and it was categoried by Khovanov and Lauda in the sequel of papers [KL09, KL11, KL10]. They treat the categoried quantum group _ U as a category with objects 2 Z and morphisms ! theQ(q)-module 1 _ U1 where the composition of 1-morphisms are given by multiplication. Therefore, the categorication of _ U is the 2-categoryU with objects indexed by the weight lattice Z, 1-morphisms are formal direct sums of composites of maps 1 : ! ,E1 : ! + 2, and F1 :! 2, and 2-morphisms are given by k-linear degree preserving combinations of tangle- like diagrams modulo local relations. This construction of the 2-categoryU(g) can be dened for any symmetrizable Kac-Moody algebra g associated to a eld k, which is shown in [KL10]. For a more detailed study of categoried quantum sl 2 and an introduction to diagrammatic algebra which is essential for studying the 2-morphism, see [Lau12]. 1.3 Odd Dierentials 1.3.1 Motivations from link homology theory Khovanov homology, categorifying a certain normalization of the Jones polynomial [Kho00, Kho02], is the simplest of a family of link homology theories associated to quantum groups and their rep- resentations. Surrounding Khovanov homology is an intricate system of related combinatorial and geometric ideas. Everything from extended 2-dimensional TQFTs [Kho00, Kho02, LP09, CMW09], planar algebras [BN02, BN05], categoryO [Str09, Str05, BS11b, BFK99], coherent sheaves on quiver varieties [CK08], matrix factorizations [KR08a, KR08b], homological mirror symmetry [SS06, CK08], arc algebras [Kho02, CK14, BS11a, BS10, BS11b, BS12], Springer varieties [Kho04, Str09, SW12], stable homotopy theory [LS14a, LS14c, LS14b], and 5-dimensional gauge theories [Wit12a, Wit12b] appear in descriptions of Khovanov homology, among many other constructions. Given the many deep connections between Khovanov homology and the sophisticated struc- tures described above, it is surprising to discover that there exists a distinct categorication of CHAPTER 1. INTRODUCTION 9 the Jones polynomial. Ozsv ath, Rasmussen, Szab o found an odd analogue of Khovanov homol- ogy [ORS13]. This odd homology theory for links agrees with the original Khovanov homology when coecients are taken modulo 2. Both of these theories categorify the Jones polynomial, and results of Shumakovitch [Shu11] show that these categoried link invariants are not equivalent. Both can distinguish knots that are indistinguishable in the other theory. The discovery of odd Khovanov homology was motivated by the existence of a spectral sequence from ordinary Khovanov homology to the Heegaard Floer homology of the double branch cover [OS05] with Z 2 coecients, see [Sza15, Blo11, KM11] for related constructions. Odd Khovanov homology was dened in an attempt to extend this spectral sequence to Z coecients rather than Z 2 . Indeed, in [ORS13] they conjecture that for a link L in S 3 , there is a spectral sequence whose E 2 term is the reduced odd Khovanov homology of L and whose E 1 term is the Heegaard Floer homology of the branched double-cover of L (with coecients in Z). A related version of this conjecture was proven in the context of instanton homology in [Sca15]. The Heegaard-Floer homology of branched double covers is closely connected with knot Floer homology which categoried the of Alexander-Conway polynomial K (q) a knot K [OS04, Ras03]. This connection between Heegaard-Floer homology and Khovanov homology is somewhat striking given that these invariants are dened in very dierent ways. However, quantum algebra sheds some light as to why such a connection is less surprising. It is well known that the Jones polynomial can be interpreted as a quantum invariant associated to the quantum group for sl 2 and its two dimensional representation. Varying the semisimple Lie algebra g and the irreducible representations coloring the strands of a link, one arrives at a whole family of quantum invariants. Later it was shown that the Alexander polynomial could be formulated as a quantum invariant, either using the quantum group associated to the super Lie algebra gl(1j1) [RS93], or using quantum sl(2) with the quantum parameter specialized to a fourth root of unity [Mur92, Mur93]. A comparison and review of these approaches appears in [Vir02]. 1.3.2 The oddication program The so called `oddication' program [LR14] in higher representation theory grew out of an attempt to provide a representation theoretic explanation for a number of phenomena observed in connection CHAPTER 1. INTRODUCTION 10 with odd Khovanov homology. This idea is that Khovanov homology shares many connections throughout out mathematics and theoretical physics, suggesting that many of the other fundamental structures connected with Khovanov homology also have odd analogs. The oddication program looks for odd analogs of structures that are often non-commutative, having the same graded ranks as traditional objects and become isomorphic when coecients are reduced modulo two. The nilHecke algebra plays a central role in the theory of categoried quantum groups, giving rise to an integral categorication of the negative half of _ U(sl 2 ) [Lau08, KL10, Rou08]. An oddication of this algebra was dened in [EKL14] which can be viewed as algebra of operators on a skew polynomial ring. The invariants under this action dene an odd version of the ring of symmetric functions [EK12, EKL14]. The odd nilHecke algebra also gives rise to \odd" noncommutative analog of the cohomology of Grassmannians and Springer varieties [LR14, EKL14]. It also ts into a 2-categorical structure [EL16, BE17b] giving an odd analog of the categorication of the entire quantum groupU q (sl 2 ) dened by Lauda in [Lau08]. In each of these cases, these structures possess combinatorics quite similar to those of their even counterparts. When coecients are reduced modulo two the theories become identical, but the odd analogues possess an inherent non-commutativity making them distinct from the classical theory. The odd nilHecke algebra appears to be connected to a number of important objects in tra- ditional representation theory. It was independently introduced by Kang, Kashiwara and Tsu- chioka [KKT16] starting from the dierent perspective of trying to develop super analogues of KLR algebras. Their quiver Hecke superalgebras become isomorphic to ane Hecke-Cliord superalgebras or ane Sergeev superalgebras after a suitable completion, and the sl 2 case of their construction is isomorphic to the odd nilHecke algebra. Cyclotomic quotients of quiver Hecke superalgebras super- categorify certain irreducible representations of Kac-Moody algebras [KKO13, KKO14]. A closely related spin Hecke algebra associated to the ane Hecke-Cliord superalgebra appeared in earlier work of Wang [Wan09] and many of the essential features of the odd nilHecke algebra including skew- polynomials appears in this and related works on spin symmetric groups [KW08a, KW08b, KW09]. CHAPTER 1. INTRODUCTION 11 1.3.3 Covering Kac-Moody algebras Clark, Hill, and Wang showed that the odd nilHecke algebra and its generalizations t into a frame- work they called covering Kac-Moody algebras [HW15, CW13, CHW13, CHW14]. Their idea was to decategorify the supergrading on the odd nilHecke algebra by introducing a parameter with 2 = 1. The covering Kac-Moody algebra is then dened overQ(q)[]=( 2 1) for certain very spe- cic families of Kac-Moody Lie algebras. The specialization to = 1 gives the quantum enveloping algebra of a Kac-Moody algebra and the specialization to =1 gives a quantum enveloping alge- bra of a Kac-Moody superalgebra. This idea led Hill and Wang to a novel bar involution q =q 1 allowing the rst construction of canonical bases for Lie superalgebras [CHW14, CW13]. In the simplest case, the covering algebraU q; can be seen as a simultaneous generalization of the quantum group _ U(sl 2 ) and the Lie superalgebra _ U(osp 1j2 ). This relationship is illustrated below. U q; _ U(sl 2 ) _ U(osp 1j2 ) !1 !1 A list of those nite and ane type Kac-Moody algebras admitting a covering algebra is shown in [HW15, Table1]. It is interesting to note the strong agreement between the existence of covering Kac-Moody algebras and the existence of an \odd link homology" predicted by the string theoretic approach to link homology constructed by Mikhaylov and Witten using D3-branes with boundary on vebrane [MW15]. The existence of a canonical basis for the covering algebraU q; led Clark and Wang to conjecture the existence of a categorication of this algebra [CW13]. The conjecture was proven in [EL16] who dened a Z Z 2 -graded categorication U of U q; . Later, Brundan and Ellis gave a simplied treatment [BE17b] using the theory of monoidal supercategories [BE17a]. This work provided a drastic simplication that makes the present work possible. Thus far, the odd categorication U of quantum sl 2 has yet to be applied to give a higher rep- resentation theoretic interpretation of odd Khovanov homology. However, the agreement between covering Kac-Moody algebras and physical theories for odd link homology is quite promising. Given the expected connections to odd link homology, the existence of spectral sequences connecting odd CHAPTER 1. INTRODUCTION 12 Khovanov homology and knot Floer homology motivates the investigation of 2-categorical dieren- tials on the odd categoried quantum group. 1.3.4 Dierential graded structures on categoried quantum group Derivations on the even categoricationU(sl 2 ) dened by Lauda were studied by Elias and Qi [EQ16a]. They were interested categorify the small quantum group for sl 2 at a (prime) root of unity. Their approach made use of the theory of Hopfological algebra initiated by Khovanov [Kho16] and devel- oped by Qi [Qi14]. The main idea in Hopfological algebra is to equip a given categorication with the structure of a p-dg algebra. This is like a dg-algebra, except that d p = 0 rather than d 2 = 0. For the theory of Hopfological algebra to work, one needsp to be a prime root of unity and the base eld that one works over should have characteristic p. Within the framework of Hopfological algebra, there have been a number of investigations into categorications at a prime root of unity. A p-DG analogs of the nilHecke algebra where studied in [KQ15]. In [EQ16a] Elias and Qi categorify the small quantum group for sl 2 at a (prime) root of unity by equipping the 2-categoryU with ap-dierential giving it the structure of a p-dg-2-category. Using thick calculus from [KLMS12], in Elias and Qi categorify an idempotented form of quantum sl 2 and some of its simple representations at a prime root of unity [EQ16b]. This involves equipping the Karoubi envelope _ U of the 2-categoryU with a p-dg structure. Related categorications studied were studied in [QS17]. Much less in known about honest dg-structures, or categorication at a root of unity working with a categorication dened over an arbitrary eld. In particular, it was shown in [EQ16a] that there are no nontrivial dierentials in characteristic zero on the original categoricationU(sl 2 ). The only clue we have is the work of Ellis and Qi that equips the odd nilHecke algebra with an honest dg-algebra structure [EQ16c] . Their work gives a categorication of the positive part of U q (sl 2 ) with q specialized to a fourth root of unity. There are a couple of points here worth highlighting. First, they work with the odd nilHecke algebra dened over an arbitrary eld or Z (no need to work in characteristic p). Second, the fourth root of unity doesn't come from considering a funny version of chain complexes withd 4 = 0; they use ordinary dg-algebras. However, the dierential they dene on the odd nilHecke algebra is not bidegree zero. Rather it has Z Z 2 -degree (2; 1) leading to CHAPTER 1. INTRODUCTION 13 so called mixed complexes, or `half graded' chain complexes of vector spaces. The eect of having mixed complexes is that the Grothendieck ring of the derived category of dg-modules over the odd nilHecke algebra takes on the structure of Z[ p 1]-algebra. So the fourth root of unity comes from the bidegree of the dierential, not from the theory of p-dg algebras. This is discussed in greater detail in section 2.2.4. Ellis and Qi suggested that their work on the dierential graded odd nilHecke algebra should extend to the odd categoried quantum group U(sl 2 ) to provide a characteristic zero lift of the dierentials dened on the original categoricationU(sl 2 ) that were studied in nite characeter- istic in [EQ16a]. Here we prove this conjecture by dening a family of dierentials on the odd 2-supercategory U. 1.3.5 Categorication Following similar arguments from [EQ16a], we show that the odd 2-category U is dg-Morita equiv- alent to a positivly graded dg-algebra enabling us to easily compute the Grothendieck ring of the dg-2-supercategory (U;@) using the theory of fantastic ltrations. As explained in Section 2.2.4, we have freedom in how we treat the Z 2 -grading in the Grothendieck group. In particular, the Grothendieck group is naturally a Z[q;q 1 ;]=( 2 1) module with [M] = [M]. We show that taking = 1 specialization results in a categorication of U q (sl 2 ) at a fourth root of unity. Taking the =1 specialization eliminates q entirely and we are left with a Z-module closely resembling gl(1j1). In particular, we have relations E 2 = F 2 = 0 and a super commutator relation for E and F . In Viro's work comparing theU p 1 (sl 2 ) andU q (gl(1j1)) Reshetikhin-Turaev functors it is shown that there is `q-less subalgebra' U 1 ofU q (gl(1j1)) that is responsible for producing the Reshetikhin- Turaev functor that is closely related to the one coming from U p 1 (sl 2 ). Both constructions were shown to have a close relationship with the Alexander polynomial. It is tempting to speculate that the appearance of U q (sl 2 ) at a fourth root of unity together with a q-less version of gl(1j1) in the theory of Alexander polynomials may be explained through this construction. We remark that there has been a number of categorications connected with gl(1j1) appearing in the literature. Positive part of gl(mj1) categoried in [Kho14, KS17]. In [EPV15] the tangle Floer CHAPTER 1. INTRODUCTION 14 dg algebra is identied with a tensor product of U q (gl(1j1)) representations and dg-bimodules were dened giving the action of quantum group generators E and F . Motivated by contact geometry, Tian dened a categocation of U q (sl(1j1)) using triangulated categories arising from the contact category of the disc with points on the boundary [Tia16, Tia14a, Tia14b]. The Alexander polynomial can be seen as the quantum link invariant associated with U q (sl(1j1)), see [KS91]. An approach to categorifying tensor powers of the vector representation of U q (gl(1j1)) based on super Schur-Weyl duality is given in [Sar16]. 1.4 A DG-extension of Symmetric Functions One of the most fundamental objects in higher representation theory is the nilHecke algebra [Lau08, Rou08, KL09]. This object is the most basic ingredient in categoried quantum groups and is intimately related to the geometry of ag varieties and Grassmannians [KK86, Lau12]. The nilHecke algebra admits a faithful action on the polynomial ring, further relating it to the combinatorics of symmetric functions and Schubert polynomials. The categorication, or higher representation theory, perspective has demonstrated that exten- sions or alternative categorications of quantum groups often have parallel implications in geometry and combinatorics. As an example, one motivation for studying the odd (spin/super) nilHecke algebra [KW08a, Wan09, EKL14, KKO13] was an attempt to supply a representation theoretic ex- planation for the appearance of \odd Khovanov homology" { a distinct link homology theory with similar properties to Khovanov homology. The odd nilHecke algebra shared many of the relationships of the usual nilHecke algebra, including connections to a new noncommutative Hopf algebra of sym- metric functions with strikingly similar combinatorics [EK12]. The odd nilHecke algebra gave \odd" noncommutative analog of the cohomology of Grassmannians and Springer varieties [LR14, EKL14]. All of these developments grew out of the discovery of an odd analog of the nilHecke algebra. Recently, Naisse and Vaz [NV16] have introduced an extension of the nilHecke algebra NH ext n that we refer to as the extended nilHecke algebra. This algebra arose in the study of a fundamental issue in higher representation theory. The problem was the fact biadjointness for E andF in the denition of categoried quantum groups [KL10, Rou08] implied that it was only possible to categorify nite dimensional modules; in particular, categorical analogs of Verma modules were CHAPTER 1. INTRODUCTION 15 inaccessible within the existing theory. Naisse and Vaz overcame this issue in the case of sl 2 , by omitting the biadjointness condition, enhancing the nilHecke algebra to the extended nilHecke algebra, and altering the main sl 2 -relation to a short exact sequence, rather than a direct sum decomposition. This work allowed for the rst categorication of Verma modules and may be an indication of the way forward in higher representation theory. Given the importance of the extended nilHecke algebra in categorifying Verma modules, this article investigates the combinatorial implications of this algebra. We dene analogs of symmetric functions ext n arising from the extended nilHecke algebra that we call extended symmetric functions. We construct families of bases for these algebras and investigate their combinatorial properties. Extending the work of Naisse and Vaz, we show that the ring ext n admits a family of dierentialsd N such that ( ext n ;d N ) is a sub-DG-algebra of the extended nilHecke algebra. Additionally, we show that the extended nilHecke algebra with its dierential d N is isomorphic to the Koszul complex associated to a regular sequence of central elements in NH n . Restricting to ( ext n ;d N ) gives a DG- algebra which is quasi-isomorphic to the cohomology ring of a GrassmannianGr(n;N). The algebra ext n has been independently discovered by Naisse and Vaz using dierent techniques [NV17b]. Our work facilitates an explicit realization of the extended nilHecke algebra NH ext n as a matrix ring of sizen! over its center, the ring of extended symmetric functions. This identies the ring ext n with the center of the DG-algebra NH ext n . The importance of the explicit isomorphism as a matrix ring over a positively graded algebra is that it allows us to dene primitive idempotents decomposing the identity 12 NH ext n . This implies NH ext n has a unique bigraded indecomposable module up to isomorphism and grading shift. Using this fact, we prove that the family of extended nilHecke algebras NH ext n , taken for all n 0, categories the bialgebra corresponding to the positive part U + (sl 2 ) of the quantized universal enveloping algebra of sl 2 , suggesting that the extended nilHecke algebra likely ts into a similar extension of KLR-algebras categorifying U + (g) for symmetrizable Kac-Moody algebras. We also dene new deformed dierentialsd N onNH ext n in section 3.5.3. The deformed dierentials also restrict to ext n and the resulting cohomology of ( ext n ;d N ) is generically isomorphic to the GL(N)-equivariant cohomology of a Grassmannian. CHAPTER 1. INTRODUCTION 16 Let us point out more clearly the relation between our work and [NV16]. In loc. cit., Vaz- Naisse dene bigraded algebras k (k2 Z 0 ) and bigraded bimodules k+1 F k , k E k+1 . These bimodules generate a 2-category which categories the Verma module for quantum sl 2 with generic highest weight. In this context, the extended nilHecke NH ext n algebra arises as the ring of bimodule endomorphisms ofF n , or equivalentlyE n . Our work provides an idempotent decomposition of E n (respectivelyF n ) as a direct sum of n! copies with shifts of a bimoduleE (n) (respectively F (n) ), thereby paving the way for a \thick calculus" version of the Vaz-Naisse 2-category, similar to what was accomplished in [KLMS12]. In this context, the ring of extended symmetric functions appears as the ring of endomorphisms ofE (n) andF (n) . It occurs that the resulting endomorphism ring is isomorphic to n , so that n+k F (n) k and kn E (n) k may be more appropriately referred to as trimodules over ( kn ; n ; k ). We remark that all of the above is compatible with the dierentials d N in the appropriate sense. See 3.5.4 for more. Finally, we mention an interpretation of the algebraic structures appearing in this subject in terms of Khovanov-Rozansky homology, both the doubly graded sl N version [KR08a] and the triply graded HOMFLY-PT version [KR08c]. The cohomology rings of GrassmannianGr(k;N) can be thought of as the sl N homology of the k-colored unknot [Wu14, Yon11], while the ring of extended symmetric functions ext k can be thought of as the HOMFLY-PT homology of the k-colored unknot [WW09]. The Koszul dierential d N considered here and in [NV16] is then a special case of Rasmussen's sl N dierential [Ras15]. We expect that the trimodules n+k F (n) k and kn E (n) k appear in this setting as the homologies of certain MOY diagrams, namely the colored theta graphs. This is likely to be related to the point of view adopted by Vaz and Naisse in [NV17a]. Chapter 2 A DG-Structure on Odd 2-category 2.1 Super dg theory Here we consider Z Z 2 -graded dg categories. This is a modest generalization of the standard theory of dg-categories, since a Z-graded dg-category induces a Z 2 -graded by collapsing the grading modulo 2. However, we not that the Z 2 grading on 2-morphisms in the 2-category U dened in section 2.4 are not the mod 2 reductions of the quantum Z-grading. It is easy to see this from the bigrading on caps and cups. We consider dierentials with respect to the Z 2 (or super) grading. If the dierential also has a nontrivial Z-grading (as is the case with the dierential on U) this can produce interesting eects on the Grothendieck ring. In particular, if the dierential has bidegree (2; 1) we are led to the notion of `half graded' complexes whose Grothendieck ring corresponds to the Gaussian integers, see section 2.2.4. The natural context for discussing Z 2 -graded dg categories is the super category formalism developed by Ellis and Brundan [BE17a, BE17b] that we review in section 2.1.1. 2.1.1 Super 2-categories Let| be a eld with characteristic not equal to 2. A superspace is a Z 2 -graded vector space V =V 0 V 1 : 17 CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 18 For a homogeneous element v2V , writejvj for the parity of v. Let SVect denote the category of superspaces and all linear maps. Note that homs Hom SVect (V;W ) has the structure of a superspace since any linear map f : V !W between superspaces decomposes uniquely into an even and odd map. The usual tensor product of|-vector spaces is again a super- space with (V W ) 0 =V 0 W 0 V 1 W 1 and (V W ) 1 =V 0 W 1 V 1 W 0 : Likewise, the tensor product f g of two linear maps between superspaces is dened by (f g)(v w) := (1) jgjjvj f(v) g(w): (2.1.1) Note that this tensor product does not dene a tensor product on SVect, as the usual interchange law between tensor product and composition has a sign in the presence of odd maps (f g) (h k) = (1) jgjjhj (fh) (gk): (2.1.2) This failure of the interchange law depending on pairity is the primary structure dierentiating super monoidal categories from their non-super analogs. If we set SVect to be the subcategory consisting of only even maps, then the tensor product equips SVect with a monoidal structure. The map u v 7! (1) jujjvj v u makes SVect into a symmetric monoidal category. We now dene supercategories, superfunctors, and supernatural transformations by enriching categories over the symmetric monoidal category SVect. See [Kel82] for a review of the enriched category theory. Denition 2.1.1. A supercategoryA is a category enriched in SVect. A superfunctor F :A!B between supercategories is an SVect-enriched functor. Unpacking this denition, the hom spaces in a supercategory are superspaces HOM A (X;Y ) = Hom 0 A (X;Y ) Hom 1 A (X;Y ) and composition is given by an even linear map. Let SCat denote the category of all (small) supercategories, with morphisms given by superfunctors. This category admits a monoidal structure. Denition 2.1.2. A 2-supercategory is a category enriched in SCat. This means that for each pair CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 19 of objects we have a supercategory of morphisms, with composition given by a superfunctor. For our purpose, it suces to consider a 2-supercategory to be an extension of the denition of a 2-category, to a context where the interchange law relating horizontal and vertical composition is replaced by the super interchange law g f Y X Y 0 X 0 = g f Y X Y 0 X 0 = (1) jfjjgj g f Y X Y 0 X 0 Eectively this means that when exchanging heights of morphisms we must take into account their parity. 2.1.2 Super dg-algebras A super dg-algebra (A;@ A ) is a superalgebra A = A 0 A 1 and an odd parity 1 |-linear map @ A : A!A satisfying d 2 and for any homogeneous a;b2A @(ab) =@ A (ab) =@ A (a)b + (1) jaj a@ A (b): A left super dg-module (M;@ M ) is a supermodule M = M 0 M 1 equipped with an odd parity |-linear map d M : M!M such that for any homogeneous elements a2A, m2M we have @ M (ma) =@ A (a)m + (1) jaj a@ M (m): If A and B are super dg-algebras, then a super dg (A;B)-bimodule is a superspace equipped with a dierential and commuting left super dg A-module and right super dg B-module structure. Denote byH(A) the homotopy category of super dg-modules given by quotienting maps of dg- modules by null-homotopies. Likewise, we denote byD(A) the derived category of dg-modules. BothH(A) andD(A) are triangulated categories. In the super setting that we are working in, the CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 20 translation functor [1] acts by the parity shift: (M[1]) k :=M k+ 1 ; @ M[1] :=@ M 2.1.3 Super dg-categories For standard results on dg-categories see [Kel06]. Denition 2.1.3. A supercategoryA is called a super dg-category if the morphism spaces between any two objects X;Y 2A are equipped with a degree 1 dierential @ @ : Hom x A (X;Y )! Hom x+ 1 A (X;Y ); which acts via the Leibniz rule @ : HOM A (Y;Z) HOM A (X;Y )! HOM A (X;Z) (g;f)7!@(gf) =@(g)f + (1) jgj g@(f): Given adg algebraA, consider the dg-enhanced module categoryA @ mod by dening the HOM- complex between two dg modules M and N to be HOM A (M;N) = Hom 0 A (M;N) Hom 1 A (M;N): The dierential @ acts on a homogenous map f2 HOM A (M;N) as d(f) :=d N f (1) jfj fd M If we takeA =| with trivial dierential then| @ mod is just the dg-category of chain complexes of super vector spaces. Denition 2.1.4. A left (respectively right) super dg-moduleM over a super dg-categoryA is a superfunctor M:A!| @ mod; resp.M:A op !| @ mod; (2.1.3) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 21 that commutes with the @-actions onA and| @ mod. 2.1.4 Super dg-2-categories Denition 2.1.5. A (strict) super dg 2-category (U;@) consists of a 2-category U, together with a dierential on 2-morphisms satisfying the super Leibniz rule for both horizontal and vertical composition. More explicitly, a super dg-2-category consists of the following data. 1. A set of objects I =;;:::, and for an ;2I we have U := Hom U (;) is a super dg-category. In particular, vertical composition of 2-morphisms obeys the super-dg- category Leibniz rule for morphisms. 2. For any pair of 1-morphisms E , E 0 in the same Hom space, the space of 2-morphisms HOM U ( E ; E 0 ) is a chain complex of super vector spaces. 3. The horizontal composition of 2-morphisms satised the Leibniz rule. That is, for any triple of objects ;;2I, then HOM U ( F ; F ) HOM U ( E ; E 0 )! HOM U ( FE ; F 0 E 0 ) (h;f)7! (hf) satises @(hf) =@(h)f + (1) jhj h@(f): CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 22 2.2 Hopfological algebra One of the primary reasons that triangulated categories are prevalent in categorication is the need to accommodate minus signs in the Grothendieck ring. For positive algebraic structures, typically additive categories suce with basis elements corresponding to indecomposable objects in the categorication. Quantum groups with their canonical basis are an excellent example of this phenomenon. However, as we expand categorication to include non-positive structures like the Jones polynomial, minus signs are lifted via the shift functor [1] for some triangulated category, with the shift functor [1] inducing the map of multiplication by1 at the level of the Grothendieck group. In his proposal for categorication at roots of unity, Khovanov showed that the traditional world of DG-categories, together with their homotopy and derived categories of modules, ts into a framework of Hopfological algebra. For our purposes, Hopfological algebra will provide a valuable perspective on the possible decategorications of graded super dg-categories. We quickly review the relevant details of Hopfological algebra needed for these purposes. For a more detailed review see [Kho16, Qi14]. 2.2.1 Basic setup LetH be a nite-dimensional Hopf algebra. Then H is also a Frobenius algebra and every injective H-module is automatically projective. Dene the stable category Hmod as the quotient of the categoryHmod by the ideal of morphisms that factor through a projective (equivalently injective) module. The category Hmod is triangulated, see for example [Hap88]. The shift functor for the triangulated structure on Hmod is dened by the cokernel of an inclusion of M as a submodule into an injective (projective) module I. We can x this inclusion by noting that for any H-module M, the tensor product H M with a free module is a free module, and the tensor product P M with a projective module is always projective [Kho16, Proposition 2]. A left integral for a Hopf algebra H is an element 2H satisfying h ="(h): Using the left integral, any H-module M admits a canonical embedding into a projective module CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 23 via M7! H M sending m7! m. Since H =|, H is a one-dimensional submodule of the free moduleH, hence it is projective. This allows us to dene a shift functor on the category of stable H-modules via T : Hmod! Hmod (2.2.1) M 7! (H=(H)) M: We now dene the basic objects of interest in the theory of Hopfological algebra that generalize dg-algebras and their modules. The reader may nd Figure 2.1 helpful for tracking the analogy. An H-module algebraB is and algebra equipped with an action ofH by algebra automorphisms. A left H-comodule algebra is an associative|-algebra A equipped with a map A : A!H A making A an H-comodule and such that A is a map of algebras. There is a natural construction to form a leftH-comodule algebra from a rightH-module algebra by forming the smash product algebra A := B#H. As a |-vector space A is just H B, with multiplication given by (h b)(` c) = X h` (1) (b` (2) )c; where we use Sweedler notation for the coproduct (`) = P (`) ` (1) ` (2) 2 H H. The left H- comodule structure onA =B#H is given by A (h b) = (h) b. LetAmod denote the category of leftA-modules and deneA H mod to be the quotient ofAmod by the ideal of morphisms that factor through an A-module of the form H N. The category A H mod is triangulated [Kho16, Theorem 1] with shift functor inherited from Hmod dened by sending an object M inA H mod to the module T (M) := (H=(k)) M: (2.2.2) Since H is a subalgebra of A = H#B, we can restrict an A-module to an H-module, which descends to an exact functor A H mod to Hmod. Dene a morphism f : M! N in A H mod to be a quasi-isomorphism if it restricts to an isomorphism in Hmod. Denote by D(B;H) the CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 24 localization ofA with respect to quasi-isomorphisms. It is shown in [Kho16, Corollary 2] and [Qi14, Corollary 7.15] thatD(B;H) is a triangulated category whose Grothendieck group is a module over K(Hmod). 2.2.2 DG-algebras from the Hopfological perspective The standard theory of dg-algebras and their modules is equivalent to the Hopfological algebra of the Z-graded Hopf super algebraH =|[D]=D 2 in the category of super vector spaces. Here deg(D) = 1 and (1) = 1 1; "(1) = 1 (2.2.3) (D) = 1 D +D 1; "(D) = 0: (2.2.4) For the super Hopf algebra|[D]=D 2 the left integral is spanned by =D. For a graded|-superalgebraB to admit anH-module structure this is equivalent to B having a degree 1 dierential d: B!B satisfying d(ab) =d(a)b(1) jaj ad(b); d 2 (a) = 0; for all a;b2B. Hence, an H-module algebra is the same thing as a dg-algebra. In a similar way, if we set A :=B#H then an A comodule algebra is the same thing as a B-dg-module. Further, one can show thatA H mod is equivalent to the homotopy category ofB-dg modules and thatD(B;H) is equivalent to the derived category of B-dg-modules. DG-algebras Hopfological algebra DG-algebra B H-module algebra DG-module A :=B#H-comodule algebra Bdgmod Amod Homotopy category of Bdgmod A H mod Derived category of B-dg-modules D(B;H) Table 2.1: Comparison between DG-algebras and Hophological algebra CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 25 2.2.3 Decategorication from the Hopfological perspective To have an interesting notion of Grothendieck group for the triangulated categories A H mod it is important that we restrict the classes of modules under consideration to avoid pathologies that can arise. In the context of Hopfological algebra the correct notion is that of compact hopfological modules from [Qi14, Section 7.2]. Denote byD c (A;H) the strictly full subcategory of compact hopfological modules inD(A;H). Denition 2.2.1 ([Qi14]). LetA be anH-module algebra over a nite dimensional Hopf algebraH over a base eld|. Dene the Grothendieck groupK 0 (D c (A;H)) to be the abelian group generated by symbols of isomorphism classes of objects inD c (A;H), modulo the relation [Y ] = [Y ] + [Z]; whenever there is a distinguished triangle insideD c (A;H) of the form X!Y !Z!T (X): Both the Grothendieck rings of categories A H mod and D(B;H) are left modules over the Grothendieck ring K 0 (Hmod) (see [Kho16, Corollary 1 and 2]). Hence, the ground ring for de- categorication provided by the theory of Hopfological algebra associated to the Hopf algebra H is determined by K 0 (Hmod). Note this group has a ring structure because Hmod has an exact tensor product. When H is quasi-triangular then K(Hmod) is commutative, so that we do not need to distinguish between left and right moduels [Qi14, Remark 7.17]. Ground ring for Grothendieck group from the Hopfological perspective In the special case when A =|, the Grothendieck group forD(|;H) is the same as Hmod since H acts trivially on| [Qi14, Corollary 9.11]. Since K 0 (A H mod) is a module over K 0 (Hmod) = K 0 (D(|;H)), the Grothendieck ring ofD(|;H) determines the ground ring for the Grothendieck group ofA H mod. In the language of dg-algebras, this just says that K 0 of the derived category of chain complexes of vector spaces determines the ground ring for K 0 of the category of dg-modules. CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 26 Consider the category of complexes of|-vector spaces. By considering the homological degree modulo two gives rise to a Z 2 grading for the dg homotopy category of (ungraded) chain complexes D(|) of vector spaces where the dierential has degree deg(d) = 1. Assuming| = Z or a eld, it follows that any complex inD(|) is isomorphic to a direct sum of indecomposable chain complexes of the following form: a single copy of| in any bidegree; a copy of S = 0!| d !|! 0 where we include the parity shift of on the right hand side to accommodate the degree of the dierential. Then the Grothendieck group is generated as a Z[]=( 2 1)-module by the symbol [|] with [|] = [|]. If the dierential d in the complex S is given by multiplication by a unit in |, then S is contractible and therefore isomorphic to 0 in K 0 (D(|)). The contractibility of S imposes the additional relation (1 +)[|] = 0: (2.2.5) The classication of objects inD(|) implies that this is the only relation, and it forces the symbol of S to be zero even when d is not multiplication by an invertible element. Hence, =1 and K 0 (D(|)) = Z[]=(1 +) = Z: (2.2.6) The homological shift|[1] is given by the cokernel of the inclusion intoH | withk7! | =D k. The injective envelope H | is two dimensional as a vector space spanned by the identity and D. We can represent H | by the complex | | D // where| includes into the right most term via the map D 1. Hence, the cokernel of this inclusion gives that|[1] =|. So we have recovered from the hopfological perspective the fact that the shift [1] is just the parity shift and at the level of the Grothendieck group we have (|[1]) = | = | = | : CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 27 We carefully reviewed the usual dg-case to set the stage for our treatment in the `mixed complex' setting. 2.2.4 Gaussian integers Consider the category of Z Z 2 graded modules. We denote byh1i a shift of the quantum (or Z-grading), and by the parity shift functor. Dene a dierential between such modules to be a map of bidegree (2; 1) that squares to 0. The main dierence between this case and the previous is that our Hopf algebra input into Hopfological algebra is now the super Hopf aglebra H =|[D]=D 2 where D has mixed degree (2; 1). A chain complex is a|-module equipped with such a dierential. Following [EQ16c] we call such complexes half-graded complexes for reasons that will become clear. Denote the corresponding homotopy category byC(|) and the derived category byD(|). Any category of Z Z 2 graded dg-modules with dierentials of bidegree (2; 1) will have a Grothendieck ring that is a module over K 0 (D(|)), so this Grothendieck ring controls the ground ring that appears in categorication via half-graded complexes. Assuming| = Z or a eld, it follows that any complex inD(|) is isomorphic to a direct sum of indecomposable chain complexes of the following form: a single copy of| in any bidegree; a copy of S = 0! b |hai d ! b+1 |ha + 2i! 0 with the rst term in any bidegree (a;b) and the right most copy in bidegree (a + 2;b + 1). Then the Grothendieck group is generated as a Z[q;q 1 ;]=( 2 1)-module by the symbol [|] with [|h1i] = q[|] and [|] = [|]. If the dierential d in the complex S is given by multiplication by a unit in|, then S is contractible and therefore isomorphic to 0 in K 0 (D(|)). For simplicity take a =b = 0 in S, the contractibility of S imposes the additional relation (1 +q 2 )[|] = 0: (2.2.7) The classication of objects inD(|) implies that this is the only relation, and it forces the symbol of CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 28 S to be zero even when d is not multiplication by an invertible element. Hence, K 0 (D(|)) = Z[q;q 1 ;]=( 2 1; 1 +q 2 ): (2.2.8) The homological shift is now given by the inclusion of| into H | via D 1 |h2i | D // so that|[1] :=|h2i and at the level of the Grothendieck group we have |[1] = |h2i = | q 2 = | since 1 +q 2 = 0. Hence, the homological shift is multiplication by1 on K 0 . If we specialize = 1, then the equation imposed by the contractible complex implies thatq 2 = 1, so the ground ring for reduces to Z. If we specialize =1 then we have the relation q 2 =1 and we get that q must be a fourth root of unity. Hence, we have the following result. Proposition 2.2.2. Given a Z Z 2 graded algebra equipped with a dierential d of bidegree (2; 1). Then the Grothendieck group associated with the category of Z Z 2 -graded dg-modules is a module over the ring Z[q;q 1 ;]=( 2 1; 1 +q 2 ): At = 1 this is just Z and at =1 this Z[ p 1]. 2.3 Results on Grothendieck groups of super dg-algebras 2.3.1 Grothendieck group of a super dg-algebra Despite our protracted discussion of Hopfological algebra, the decategorication of categories of super dg-modules is not so unlike the decategorication of normal dg-modules. We detoured through Hopfological algebra to highlight the fact that the Grothendieck ring will have the structure of a CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 29 module over the Gaussian integers Z[ p 1]. Just as in the usual theory of dg-modules over a dg- algebra A, to have a sensible notion of Grothendieck group ofD(A), we pass to the compact or perfect derived categoryD c (A). The categoryD c (A) is a subcategory ofD(A) consisting compact dg modules, that is, those super dg modules M such that the functor HOM D(A) (M;) commutes with innite direct sums. For our purposes the connection between compact dg modules and nite- cell modules will be of particular relevance. See for example [EQ16a, Example 2.4]. The Grothendieck groupK 0 (A) of a dg algebraA is the quotient of the free abelian group on the isomorphism classes [M] of compact dg-modules M by the relation [M] = [M 1 ] + [M 2 ] whenever M 1 !M!M 2 !M 1 [1] is an exact triangle of compact objects inD(A). 2.3.2 Positively graded dg-algebras A Z-graded dg-algebra is called a positive dg-algebra (see [Sch11]) if it satises the following 1. the algebra A = i2Z A i is non-negatively graded, 2. the degree zero part A 0 is semisimple, and 3. the dierential acts trivially on A 0 . The calculation of the Grothendieck ring of a positively graded dg-algebra is greatly simplied. Theorem 2.3.1 ([Sch11] and [EQ16a] Corollary 2.6). Let A be a positive dg algebra, and A 0 be its homogeneous degree zero part. Then K 0 (A) =K 0 (A 0 ): 2.3.3 Grothendieck ring of super dg-2-categories Denition 2.3.2. For a DG 2-category (U;@) dene the homotopy and derived categories as C(U) := M ;2I C( U ); D(U) := M ;2I D( U ): (2.3.1) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 30 The corresponding Grothendieck rings are dened via direct sums of the hom DG-categories K 0 (U) := M ;2I K 0 ( U ): (2.3.2) 2.3.4 Fantastic ltrations In this section, we give a review of the fantastic ltration and recall the related theorems from [EQ16a]. Fantastic ltration are an essential tool in this work for determining the Groethendieck ring of the odd dg 2-category U. The key issue is that if A is a dg-algebra the direct sum decomposition ofA-modules does not necessarily commute with the dierential. However, if there exists a fantastic ltration F on an A-module Ae, where e is an idempotent, then the direct sum decomposition of Ae as A-modules becomes a direct sum decomposition of dg-modules. We collect several important results on fantastic ltrations from [EQ16a, Section 5] that are easily adapted to the super dg-setting. Lemma 2.3.3. Let R be a ring and the elements u i ;v i 2R, where i2I is a nite set, satisfy the following conditions: u i v i u i =u i (2.3.3) v i u i v i =v i (2.3.4) v i u j = i;j (2.3.5) then e = P i u i v i is an idempotent and we have a direct sum decomposition Re = i Rv i u i . Note that u i v i is an idempotent for each i2 I, as u i v i u i v i = u i v i , and moreoverfu i v i g i2I is a set of orthogonal idempotents, as for any i6=j, u i v i u j v j =u j v j u i v i = 0 and it follows that e is an idempotent and Re = i Rv i u i . For a dg-algebra A and any idempotent e2 A, the A-module Ae is an A @ mod summand if for any a2A, we have v for any b2A. By the Leibniz rule, @(abe) =@(a)be + (1) jaj a@(b)e + (1) jaj+jbj ab@(e) =@(ab)e + (1) jaj+jbj ab@(e) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 31 so that@(abe)2Ae if@(e) = 0. The computation of the dierential of an idempotente is important for determine if Ae is compact in the derived categoryD(A), since @(e) = 0 implies that Ae is cobrant and has a compact image inD(A). Proposition 2.3.4. Let (A;@) be a super dg-algebra, i2I a nite index set, u i ;v i 2A satisfying the hypothesis of Lemma 2.3.3. Suppose thate = P i u i v i , and< is a total order onI. AnI-indexed super A-module ltration F of Ae is dened by F i := X ji Ru j v j and F ; := 0, so that F i =F i =Av i u i as A modules. Then the following conditions are equivalent: 1. F is a ltration by super dg-modules, so that Av i u i is a super dg-module and the subquotient isomorphism is an isomorphism of super dg-modules. 2. The following equations are satised for all i2I, v i @(u i ) (2.3.6) u i @(v i )2F <i (2.3.7) Proof. We rst give the proof (2) implies (1) since it is more interesting, and converse follows easily. (2) =) (1): In order to show that Av i u i is a dg A-module we check that Av i u i is @ closed. It suces to check @(v i u i )2Av i u i , @(v i u i ) =@(v i )u i + (1) jvij v i @(u i ) =@(v i )u i =@(v i )u i v i u i 2Av i u i and for any a2 A, it implies @(av i u i ) = @(a)v i u i + (1) jaj a@(v i u i )2 Av i u i . Next we show that for each i inI, F i is @ is dg A-module, which we prove using induction on i. If i = 0, then @(u 0 v 0 ) = @(u 0 )v 0 + (1) ju0j u 0 @(v 0 ) = @(u 0 )v 0 u 0 v 0 , since u 0 @(v 0 ) = 0. Then assume the result is CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 32 true for all j <i and show it for i. @(u i v i ) =@(u i )v i + (1) juij u i @(v i ) @(u i )v i u i v i + (1) juij u i @(v i )2Ru i v i +F <i F i : For the last part, we have F i =F <i = Av i u i as A-modules, to show that this is an isomorphism of dg A-modules, we need to show map the module isomorphism commute with the @. Let f ui : F i =F <i ! Av i u i be the multiplication map on the right by u i and as an A-module F i =F <i = Au i v i , @(f ui (ru i v i )) =@(ru i v i u i ) =@(r)u i v i u i + (1) jrj r@(u i )v i u i + (1) jrj+juij ru i @(v i )u i + (1) jrj+juij+jvij ru i v i @(u i ) =@(ru i v i )u i =f ui @(ru i v i ) rst equality in the last line follows from v i @(u i ) = 0. Let f vi :Av i u i !F i =F <i be the multipli- cation map on the right by v i , then @(f vi (rv i u i )) =@(rv i u i v i ) =@(r)v i u i v i + (1) jrj r@(v i )u i v i + (1) jrj+jvij rv i @(u i )v i + (1) jrj+jvij+juij rv i u i @(v i ) f vi (@(rv i u i )) + (1) jrj+jvij+juij rv i u i @(v i ) since (1) jrj+jvij+juij rv i u i @(v i )2F <i , @ commutes with f vi . (1) =) (2): For the converse, since Av i u i is dg A-module, it is closed under @ which means @(v i u i )2 Av i u i and it implies v i @(u i )2 Av i u i . If it is nonzero then v i @(u i ) = v i au i v i for some nonzero a2A. Then it contradicts with f ui is a dg A-module map 0 =@(f ui (ru i v i ))f ui (@(ru i v i )) = (1) jrj+juij+jvij ru i v i @(u i ) = (1) jrj+juij+jvij ru i v i av i u i Therefore, v i @(u i ) = 0. Similarly, we can show that u i @(v i )2 F <i by using F i =F <i = Au i v i dg-module map. CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 33 Denition 2.3.5. If the ltration F in Proposition 2.3.4 satises @(e) = 0 and @(v i u i ) = 0 for all i2I, then it is called a fantastic ltration on the dg-module Ae. The main advantage of the fantastic ltration is that it gives a direct sum decomposition of the images of idempotents as dg-modules. Theorem 2.3.6. Let A be a dg superalgebra, fu i ;v i g i2I a nite set of elements of A satifying Proposition 2.3.4, then there is a fantastic ltration on the dg module Ae if and only if there exists a total order on I such that v i @(u j ) = 0 forji. Moreover, in K 0 (A), we have the relation [Ae] = X i2I [Av i u i ]: Proof. We leave the "if" direction as an exercise and for the "only if" direction, suppose that we have v i @(u j ) = 0 for ji and we need to show that the conditions in Proposition 2.3.4 is satised. v i @(u j ) = 0 follows by the assumption when j =i. We have v i u j = 0 whenever i6=j, then 0 =@(v i u j ) =@(v i )u j + (1) jvij v i @(u j ) By the assumption v i @(u j ) = 0 for j > i, therefore we have @(v i )u j = 0. Note that u i @(v i )2 F <i is satised if and only if u i @(v i )u j v j = 0 for j > i, and with @(v i )u j = 0 then u i @(v i )u j v j = 0 follows. 2.4 The odd 2-category for sl(2) 2.4.1 The odd nilHecke ring The odd nilHecke ringONH a to be the graded unital associative ring generated by elementsx 1 ;:::;x a of degree 2 and elements @ 1 ;:::;@ a1 of degree2, subject to the relations @ 2 i = 0; @ i @ i+1 @ i =@ i+1 @ i @ i+1 ; (2.4.1) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 34 x i @ i +@ i x i+1 = 1; @ i x i +x i+1 @ i = 1; (2.4.2) x i x j +x j x i = 0 (i6=j); @ i @ j +@ j @ i = 0 (jijj> 1); (2.4.3) x i @ j +@ j x i = 0 (i6=j;j + 1): (2.4.4) For each w 2 S a , choose a reduced expression w = s i1 s i ` in terms of simple transpositions s i = (i i + 1). Dene @ w =@ i1 @ i ` : (2.4.5) 2.4.2 The odd categoried quantum group In [BE17b] Ellis and Brundan give a minimal presentation of the 2-category U that requires the invertibility of certain maps. Here we give a more traditional presentation by including the additional relations on 2-morphisms that equivalent to the invertibility of the maps. Let ji;j :=jij (hi;i + 1) (2.4.6) wherejij = 1 and for sl 2 , we haveji;j = + 1. The odd 2-supercategory U = U(sl 2 ) is the 2-supercategory consisting of objects for 2 Z, for a signed sequence " = (" 1 ;" 2 ;:::;" m ), with " 1 ;:::;" m 2f+;g, dene E " :=E "1 E "2 :::E "m whereE + :=E andE :=F. A 1-morphisms from to 0 is a formal nite direct sum of strings E " 1 hti = 1 0E " 1 hti for any t2 Z and signed sequence " such that 0 = + 2 P m j=1 " j 1. CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 35 2-morphisms are Z-modules spanned by (vertical and horizontal) composites of identity 2- morphisms and the following tangle-like diagrams OO +2 :E1 hti!E1 ht + 2i OO OO :EE1 hti!EE1 ht 2i (parity 1) (parity 1) JJ : 1 hti!FE1 ht + 1 +i TT : 1 hti!EF1 ht + 1i (parity 0) (parity ( + 1)) WW :FE1 hti! 1 ht + 1 +i GG :EF1 hti! 1 ht + 1i (parity ( + 1)) (parity 0) for every ;t2 Z. The degree of a 2-morphism is the dierence between degrees of the target and the source. Diagrams are read from right to left and bottom to top. The rightmost region in our diagrams is usually colored by . The identity 2-morphism of the 1-morphismE1 is represented by an upward oriented line (likewise, the identity 2-morphism ofF1 is represented by a downward oriented line). The fact that we are dening a 2-supercategory means that diagrams with odd parity skew commute. The 2-morphisms satisfy the following relations (see [BE17b] for more details). 1. (Odd nilHecke) TheE's carry an action of the odd nilHecke algebra. Using the adjoint structure this induces an action of the odd nilHecke algebra on theF's. OO OO = 0; OO OO OO OO OO OO = OO OO OO OO OO OO (2.4.7) OO OO = OO OO + OO OO = OO OO + OO OO (2.4.8) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 36 2. (Right adjunction axioms) OO OO 2 = OO 2 + 2 = OO + 2 (2.4.9) 3. (Parity left adjoint) OO 2 = (1) +1 OO OO 2 OO + 2 = + 2 (2.4.10) 4. (Bubble relations) Dotted bubbles of negative degree are zero, so that for all m 0 one has m = 0 if m< 1, m = 0 if m< 1. (2.4.11) Dotted bubble of degree 0 are equal to the identity 2-morphism: 1 = Id 1 for 1, 1 = Id 1 for 1. We use the following notation for the dotted bubbles: +m := m+1 ; +m := m1 ; so that deg +m ! = deg +m ! = 2m: CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 37 The degree 2 bubbles are given a special notation as follows: N := 8 > > > > > < > > > > > : +1 = ; 0, +1 1 = ; 0. (2.4.12) We call a clockwise (resp. counterclockwise) bubble fake if m +n 1 < 0 and (resp. if mn 1 < 0). The fake bubbles are dened recursively by the homogeneous terms of the equation X r;s0 r+s=t +2r +2s = t;0 : (2.4.13) +2n+1 = +2nN ; +2n+1 = +2n N (2.4.14) 5. (Centrality of odd bubbles) By the super interchange law it follows that the odd bubble squares to zero. Further, we have OO N = OO N N = N (2.4.15) 6. (Cyclicity propeties) OO = 2 N OO (2.4.16) The cyclic relations for crossings are given by = OO OO OO OO = OO OO OO OO (2.4.17) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 38 Sideways crossings satisfy the following identities: OO := OO OO OO = OO OO (2.4.18) OO := (1) +1 OO OO OO = OO OO (2.4.19) 7. (Odd sl(2) relations) OO = OO + X f 1 +f 2 +f 3 =1 (1) f2 OO f1 + f2 f3 ; (2.4.20) OO = OO + X f 1 +f 2 +f 3 =1 (1) f2 OO f1 + f2 f3 : 2.4.3 Additional Relations For later convenience we record several relations that follows from those in the previous section, see [BE17b] for more details. 1. (Dot Slide Relations) OO n = (1) b n 2 c OO n n = (1) b n 2 c n (2.4.21) (1) b n 2 c OO n = 8 > > > > > > > > < > > > > > > > > : OO n if n is even (1) OO n + 2 OO n1 N if n is odd (2.4.22) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 39 (1) b n 2 c n = 8 > > > > > > > < > > > > > > > : n if n is even (1) n + 2 n1 N if n is odd (2.4.23) 2. (Bubble Slide Relations) OO +n = X r0 (2r + 1) OO 2r +n2r (2.4.24) OO +n = X r0 (2r + 1) OO 2r +n2r (2.4.25) 3. (Pitchfork Relations) OO = OO = (2.4.26) OO OO = OO OO OO = OO (2.4.27) OO = OO OO OO = OO OO (2.4.28) OO = OO = (2.4.29) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 40 4. (Curl Relations) For all n 0 we have, OO n = n X r=0 (1) (r+1) OO r +nr ; OO n = n++2 X r=0 (1) r OO r ++nr n ; (2.4.30) Proposition 2.4.1. We deduced the following relations for the odd 2-category U using a bending trick from the curl relations (2.4.30). OO = X r=0 (1) (+r+1) NN r +(r) (2.4.31) = X r=0 (1) (+r) (r) +r (2.4.32) 2.5 Derivations on the odd 2-category In this section, we study derivations on the odd 2-category U = U(sl 2 ). We will dene and then give a classication of derivations on the odd 2-category U. A derivation on a super-algebraA is an endomorphism@ which is of degree 2 and satises the Leibniz rule, so for any homogeneous a;b2A, @(ab) =@(a)b + (1) jaj a@(b) wherejaj is the parity of the element a. A derivation @ is called a dierential on A if @ 2 = 0. Denition 2.5.1. A derivation@ on the Z Z 2 -graded odd 2 categoryU is a (categorical) derivation of bidgree (2; 1) on the space of 2-morphisms which satises the Leibniz rule for both horizontal and vertical multiplication of 2-morphisms. The reason that we take the derivation to be of degree 2 is due to the diagrammatic calculus, since CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 41 each dot increases the degree of the 2-morphism by 2. We start by dening @ on each generating 2-morphism as an arbitrary linear combination of 2-morphisms of degree two higher. We derive constraints on the arbitrary coecients by requiring the derivation@ to satisfy the dening relations of U. Therefore using the basis theorem for the space of 2-morphisms, the most general form of a potential derivation on U of bidgree (2; 1) has the form: @ OO +2 ! := 1; OO +2 2 + 2; OO +2 N + 3; OO +2 +2 (2.5.1) @ OO OO := 1 ; OO OO + 2 ; OO OO (2.5.2) + 3 ; OO OO + 4 ; OO OO N (2.5.3) @ := a 2 + b 2 N (2.5.4) @ PP := a PP + b NN N (2.5.5) @ :=c + d N (2.5.6) @ NN := c 2 NN + d 2 NN N (2.5.7) Derivation of more complicated diagrams can be found by using dening relations of the odd 2-category _ U(sl 2 ) and the Leibniz rule. CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 42 The derivation @ respects the dening relations of the odd 2-category U. For example, let us consider the right adjunction axiom (2.4.9), derivation @ respects this relation means that if you take @ of both sides then @ of right hand side and @ of the left hand side will agree. Therefore, in order to understand the relationship between coecients in the denition of the derivation @ above, we study @ of the dening relations of the odd 2-category U. Let us recall the right adjunction relation below: OO a b OO +2 = OO +2 (2.5.8) Derivation@ of the left hand side is calculated using the Leibniz rule, as it is a composition ab of a and b, which is denoted as ab for short, so @(ab) = @(a)b + (1) jaj @(b), and the parity of a is even, sojaj = 0. Diagrammatically we have @ 0 B B @ OO OO +2 1 C C A = (a + a ) OO +2 + (b + b ) OO N +2 (2.5.9) and derivation of the identity 2-morphism is zero, which can be deduced rst realizing identity is composition of two identity 2-morphisms and using the Leibniz rule, so we have @ 0 B @ OO +2 1 C A = 0 then using the linear independence of the 2-morphisms in (2.5.9), we obtain the following equations CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 43 from the right adjunction relation: a =a b =b Similarly, we take @ of left adjunction relation (2.4.10) 0 = (1) +1 @ 0 B @ OO +2 1 C A = @ 0 B B @ OO OO +2 1 C C A (2.5.10) = (c + (1) +1 c ) OO +2 + (d + d ) OO N +2 (2.5.11) and it gives c = (1) c d =d Next, we move on to the odd Nilhecke relations, and rst we take @ of the (2.4.7). 0 =@ 0 @ OO OO 1 A = 1 ; OO OO + 2 ; OO OO + 3 ; OO OO + 4 ; OO OO N 1 ; OO OO 2 ; OO OO 3 ; OO OO 4 ; OO OO N = ( 2 ; 3 ; ) OO OO CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 44 which implies 3; = 2; then we update the derivation of the upward crossing as: @ OO OO = 1 ; OO OO + 2 ; OO OO 2 ; OO OO + 3 ; OO OO N then we take derivation of the second odd Nilhecke relation (2.4.8) and use the fact that @ of the identity 2-morphism is zero. @ OO OO + @ OO OO = 0 @ OO OO = 1;+2 OO OO 2 + 2;+2 OO OO N + 3;+2 OO OO +2 1 ; OO OO 2 ; OO OO 2 + 2 ; OO OO 3 ; OO OO N = 1;+2 OO OO 2 2;+2 OO OO N + 3;+2 OO OO +2 + 3 3;+2 OO OO 2 1 ; OO OO 2 ; OO OO 2 CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 45 + 2 ; OO OO 3 ; OO OO N where the last equality follows from bubble slide relation (2). @ OO OO = 1 ; OO OO + 2 ; OO OO 2 ; OO OO + 3; OO OO N 1; OO OO 2 2; OO OO N 3; OO OO +2 = 1 ; OO OO + 2 ; OO OO 2 ; OO OO 2 2 ; OO OO 2 ; OO OO 3 ; OO OO N + 3; OO OO N 1 ; OO OO + 1 ; OO OO 1; OO OO 2 2 ; OO OO N + 2; OO OO N 3; OO OO +2 where the last equality follows from pushing all the dots through crossing using odd NilHecke relation (2.4.8). Then we have 0 =@ OO OO + @ OO OO = ( 1;+2 1; 2 2; ) OO OO 2 + 3 3;+2 OO OO 2 CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 46 + ( 2;+2 + 2; ) OO OO N + ( 3;+2 3; ) OO OO +2 + ( 1; + 2; + 1; ) OO OO + ( 1; 2; 1; ) OO OO ( 3; + 2; ) OO OO N then we get the following set of equations: 1;+2 1; 2 2; = 0 3;+2 3; = 0 3;+2 = 0 2;+2 2; = 0 3;+2 3; = 0 3; + 2; = 0 1; + 2; + 1; = 0 From which we can deduce that, 3 := 3; doesn't depend on weight and 3 = 0 (2.5.12) and 2; does not depend on the weight as well. 2 := 2; = 2;+2 (2.5.13) If we combine the rst and the last equations we get 2 1; = 1;+2 1; (2.5.14) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 47 The equation (2.5.14) above is what we get from also taking@ of second Nilhecke relation of (2.4.7), and we omit the proof due to its length. Now, we use the equation we get from checking @ of the above relations and update the denition of the derivation @ : @ OO +2 ! = 1; OO +2 2 + 2 OO +2 N (2.5.15) @ OO OO = 1 ; OO OO + ( 1; 1; ) OO OO (2.5.16) + ( 1; 1; ) OO OO 2 OO OO N (2.5.17) @ = a 2 + b 2 N (2.5.18) @ PP =a PP b NN N (2.5.19) @ =c + d N (2.5.20) @ NN = (1) c 2 NN d 2 NN N (2.5.21) with the equation 2 1; = 1;+2 + 1; CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 48 By induction on the number of dots and using Leibniz rule, for n 2 we have @ 0 B @ OO +2 n 1 C A = 1; n;odd OO n+1 +2 + (1) n+1 n 2 OO n +2 N (2.5.22) Downward dot is dened by the following relation 2 := OO (2.5.23) then the derivation @ of the downward dot is @ 0 B @ 2 1 C A = (2a 2 1; ) 2 2 + 2 2 N (2.5.24) Remark 2.5.2. Note that for any 2-morphisms a;b;c, @(abc) is well-dened, which means that it is independent from the order we choose to apply the Leibniz rule. @(abc) =@(a(bc)) = (@(a))bc + (1) jaj @(bc) = (@(a))bc + (1) jaj a(@(b)c + (1) jbj @(c)) =@(a))bc + (1) jaj a@(b)c + (1) jaj+jbj ab@(c) =@(ab)c + (1) jaj+jbj ab@(c) =@((ab)c) For n 2, using induction on the number of dots we get @ 0 B @ n 2 1 C A = (2a 1; ) n;odd 2 n+1 + (1) n+1 n 2 n 2 N (2.5.25) Since the rest of the relations include bubbles in them, we study the derivation @ of an n- labeled bubble, i.e., degree 2n-bubble by rst realizing it as a vertical composition of three 2-morphisms CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 49 abc and then by using the Leibniz rule. Figure below illustrates this description. a b c ⇤ +n Figure 2.1: Decomposition of an n-bubble for taking derivation Remark 2.5.3. In the computations, we rst consider real bubbles, which are positively dotted bubbles. The reason is that we take the derivation by decomposing the bubble into generating 2- morphisms, and we don't have a denition of derivative of a negative dotted 2-morphisms. However, derivation of the fake bubbles can be computed by derivation of real bubbles via the odd Grass- mannian relation (2.4.13). We will later see that the same formula will apply to fake bubbles as well. If we use the decomposing argument as described in the gure above then the derivation of odd bubble is computed as follows: @ N ! = 8 > > > > > > < > > > > > > : (a 2 +c 2 + 1;2 ;odd ) +2 if 0 (a +c + 1;2 ;odd ) +2 if 0 (2.5.26) There is another approach nding @ of an odd bubble. Assume that 0 so that we have a real bubble of label n, and take n 1. Its derivation can be expressed as a linear combination of a real bubble of label n + 1 and a product of two bubbles one is of label n and the other one is of label 1, namely the odd bubble. @ +n ! = x n; +n+1 + y n; +n N (2.5.27) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 50 then for n = 1, we have @ N ! = x 1; +2 + y 1; N N = x 1; +2 where the last equality follows from N 2 = 0. If we use the centrality of odd bubbles (2.4.15) we get @ 0 B @ OO N 1 C A = @ 0 B @ OO N 1 C A x 1;+2 OO +2 = x 1; OO +2 then we slide the label 2 bubble on the left hand side through the identity 2-morphism using the bubble slide relation (2), we get: (x 1;+2 x 1; ) OO +2 + 3x 1;+2 OO = 0 From which we deduce that x 1 does not depend on the weight and is zero. x 1 :=x 1; =x 1;+2 = 0 (2.5.28) This observation implies that the @ of an odd bubble is zero, and if we combine it with (2.5.26), it implies a +c + 1; ;odd = 0 (2.5.29) and same result is true for 0. At this point we would like to clarify that real odd bubble is equal to the fake odd bubble using the relations of odd 2-category U. CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 51 Proposition 2.5.4. For all 2 Z, we have the the equality of real and fake odd bubbles. +1 = +1 Proof. First we assume that 0 and recall the equation (5.8) from [BE17b], for all t> 0 X r;s2Z r+s=t2 (1) s +(+1+s) +(1+r) = 0 If we choose t = 1, X r;s2Z r+s=1 (1) s +(+1+s) +(1+r) = 0 then we have nonzero terms in the sum above if s 1 r 1 then keeping in mind that r +s =1 we have 0 = (1) +1 + (1) +1 +1 and the result follows. Similarly, one can show that the same holds for < 0. We can use odd innite Grassmannian relation (2.4.13) and (2.4.14) to express fake bubbles in terms of the real bubbles to compute their derivation. Proposition 2.5.5. Derivation of an odd labeled bubble is zero. For n 0, @ +2n+1 ! = @ +2n+1 ! = 0 (2.5.30) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 52 Proof. The proof of the statement follows easily using the relation (2.4.14) and the Leibniz rule. Let us use the notation as in (2.5.27), for 0 @ +2n+1 ! = @ 0 B @ +2n N 1 C A = x 2n; +2n+1 N + y 2n; +2n N N = 0 where the last equality follows from the relation (2.4.14) and using N 2 =. Proof of < 0 follows similarly. Dierential of n-labeled real bubbles, for n 2 can be computed using Leibniz rule as in (2.5.26) For 0, we have: @ +n ! = n;even (a 2 + 1;2 ;even c 2 +b 2 (n + 1) 2 + (1) d 2 ) +n+1 (2.5.31) For 0, we have: @ +n ! = n;even (a +c + (1) +1 1; ;even b (n 1) 2 + (1) +1 d ) +n+1 (2.5.32) Derivation of degree 0-bubble is zero also implies relations on the coecients in the denition of the derivation @. For 1 0 =@ +0 = (a 2 +b 2 + 1;2 ;even ( 1) 2 c 2 + (1) d 2 ) N (2.5.33) and for 1 0 =@ +0 = (c a b + (1) +1 1; ;even + ( + 1) 2 (1) d ) N (2.5.34) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 53 and we have 8 > > > > > > < > > > > > > : a 2 +b 2 + 1;2 ;even ( 1) 2 c 2 + (1) d 2 = 0 if 1. c a b + (1) +1 1; ;even + ( + 1) 2 (1) d if 1. (2.5.35) Then we combine the equations(2.5.35) with (2.5.33) and (2.5.34) and prove the following statement Proposition 2.5.6. @ +n ! = n;even n 2 +n+1 , for 0 (2.5.36) @ +n ! = n;even n 2 +n+1 , for 0 (2.5.37) Next, we take the derivation of the odd dot cyclicity relation (2.4.16). @ 0 B B @ OO 1 C C A = 2@ 0 B @ N 1 C A @ 0 B B @ OO 1 C C A and it is equal to (2a 1; ) 2 + 2 N = (2c + (1) +1 1; ) 2 + 2 N then it gives 2c + (1) +1 1; =2a 1; which can be expressed as c =a ;odd 1; (2.5.38) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 54 If we combine (2.5.38) with the equation (2.5.29) obtained from@ of degree-0 bubble is zero, we can express d as d = (1) +1 (2a + 1; +b ( + 1) 2 ) (2.5.39) and it is satised for all . By using (2.4.18) and (2.4.19) we can compute the derivation of @ of side crossings. For the compu- tation we use Leibniz rule and odd Nilhecke relations. @ OO = (a a 2 1; + 1; ) OO + ( 1; 1; ) OO + (b b 2 2 ) OO N + ( 1; +a ) PP @ OO = ( 1;2 1;2 ) OO + (c c 2 + (1) +1 ( 1;2 1;2 )) OO + (2(c 2 c ) + (1) (d d 2 ) 2 + (1) ( 1;2 1;2 )) OO N + ((1) +1 1;2 +c 2 ) NN Similarly, using the relation (2.4.17) we can compute the derivation of the downward crossing as well. @ = (a 2 +a 4 1;2 + 1;2 ) + (a 2 a 4 1;2 + 1;2 ) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 55 2 N + ( 1;2 +a 2 +a 4 ) So far derivation @ we dene on the 2-morphisms of the odd 2-category U satises the right and left adjunction relations, all of three odd Nilhecke relations, and odd dot cyclicity relation. We also observed that by using the centrality of the odd bubble, @ of an odd bubble gives some relations between the coecients in the denition of the derivation @, and furthermore we used the fact that derivation of degree-0 bubble is zero and got some additional equations for the coecients. The next relation we need to impose is the oddsl(2) relations (2.4.20), however with the current state of the denition of the derivation@, it would be quite complicated to take the@ of the relation (2.4.20). Our main goal with this work is to dene a dierential @ on the odd 2-category U which gives rise to a fantastic ltration so that we can take the Groethendieck ring K 0 , we will consider the derivation which satises the above listed relations is the most general form of a derivation we can dene on the space 2-morphisms of the odd 2-category U. Therefore, we proved the following Proposition 2.5.7. Any derivation@ on the space of 2-morphisms of the odd 2-category U is of the following form on the generators: @ OO +2 ! = 1; OO +2 2 + 2 OO +2 N (2.5.40) @ OO OO = 1 ; OO OO + ( 1; 1; ) OO OO (2.5.41) + ( 1; 1; ) OO OO 2 OO OO N (2.5.42) @ = a 2 + b 2 N (2.5.43) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 56 @ PP =a PP b NN N (2.5.44) @ = (a 1; ;odd ) + d N (2.5.45) @ NN = (1) (a 2 1; 2;odd ) NN d 2 NN N (2.5.46) and 2 1; = 1;+2 + 1; (2.5.47) d = (1) +1 (2a + 1; +b ( + 1) 2 ) (2.5.48) There are two ways to proceed after this denition. The rst one is, we can continue computing the derivation of the dening relations of the odd 2-category U and obtain more equations which gives relations between coecients in the denition of the derivation@. Then we declare that@ 2 = 0 in order to obtain a dierential and by computing @ 2 = 0 of each generator we can nd the nal denition of the dierential on U. However, after we equip the odd 2-category U with a dierential @, the next step is to take the Grothendieck ring K 0 of this odd DG 2-category U and decategorify it. This is where things get complicated, since the question of nding Grothendieck ring K 0 of an arbitrary DG-category is a very hard problem and not known most of the time. What we can do is to use some of the known tools to compute the Grothendieck ring, and we use the ones studied in [EQ16a, EQ16c] as the second way. The essential tool we use is the fantastic ltration dened on the dg-modulesAe for any idempotente, which implies the idempotent decomposition of e gives isomorphism Ae = P i u i v i not only as A-modules but as dg-modules. CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 57 2.5.1 Derivations on odd 2-category arising from the fantastic ltration Our main goal is to classify the derivations on the odd 2-category U arising from fantastic ltrations. We refer the reader to Section 2.3.4 for the preliminaries on the Fantastic ltration on A-modules for a dg-algebra A. First, we need to check that the datafu i ;v i g i2I from which we built the fantastic ltration satises conditions in Lemma 2.3.3 with@(e) = 0 and@(v i u i ) = 0. Then we need to checkv i @(u j ) = 0 for ji in order to have a fantastic ltration on the dg-module Ae. Let us consider the case when is positive and use the Denition 2.13 of [BE17b] to choose u n and v n . u n := X r0 (1) (+n+r+1) NN r nr2 , for 0n 1 u := OO v n := n , for 0n 1 v := OO Below we show that u n and v n satisfy the rst set of requirements to have the Fantastic ltration: @(u n v n ) = 0 for all 0n v s u t = 0, for s6=t: CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 58 For 0n 1 v n u n = X r0 (1) (+n+r+1) n+r nr2 = X r0 (1) (+n+r+1) (n+r+1)+ (nr1)+ = (1) 0+ 0+ = (1) 1 We know that negative degree bubbles are zero, so in order to have nonzero terms in the above sum, for the clockwise bubble we must have rn 1 and for the counterclockwise bubble n 1r which gives r =n 1 and the result follows. Since dierential of the identity 2-morphism is zero, we have @(v n u n ) = 0, for 0n 1. Also note that for n =, @(v u ) = @ 0 @ OO OO 1 A = @ OO ! = 0 Next, we check that v n u m = 0 for n6=m, and we do it by rst verifying it for 0n;m 1 v n u m = X r0 (1) (+m+r+1) n+r mr2 = X r0 (1) (+n+r+1) (n+r+1)+ (mr1)+ (2.5.49) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 59 Only positive degree bubbles survive in the sum, so we must have m 1rn 1 If m > n then above inequality is not satised and there is no n;m value giving nonzero terms in the sum (2.5.49) and v n u m = 0 follows easily. On the other hand, if n<m then by using relation (2.4.13) and (2.4.14) one can deduce that the condition v n u m = 0 is satised as well. Now, let us verify that v u n = 0 for 0n 1 v u n = X r0 (1) (+n+r) OO r nr2 = X r0 0 B B @ (1) n OO r nr2 + (1) (n+1) X s;t0;s+t=r1 (1) t OO r nr2 1 C C A = X r0 X s;t0;s+t=r1 (1) (n+t+1) PP s t nr2 = X r0 X s;t0;s+t=r1 (1) (n+t+1) PP s (+t+1) (+nr1) where the second equality comes from (3.3) in [BE17b] and the third equality follows since the rst term in the second line is zero by (5.17) in [BE17b]. Only the positive degree bubbles survive in the sum, so we must have 1t<rn 1 Since n is nonnegative, there is no positive s;t;r values giving positive degree bubbles, therefore CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 60 v u n = 0. Finally, v n u = 0 for 0n 1 follows from (2.13) in [BE17b] v n u = n = 0 These computations verify that our choice of u n and v n satises the hypothesis of the Fantastic ltration theorem. By the Theorem 2.3.6, in order to have a fantastic ltration we want the following condition satised, for some total order on I and i;j2I, v i @(u j ) = 0, for ji. (2.5.50) We will divide the computation into cases: First we consider i = j = and using the denition of the dierential we compute v @(u ) and set it equal to zero. 0 = v @(u ) = ( 1;2 1;2 ) OO OO (c c 2 + (1) +1 ( 1;2 1;2 )) OO OO (2c + (1) d + 2(1) ( 1;2 1;2 ) 2 + (1) +1 d 2 + 2c 2 ) OO OO N ((1) +1 1;2 +c 2 ) OO r CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 61 then v @(u ) = 0 gives the following equations 1;2 1;2 = 0 c c 2 + (1) (+1) ( 1;2 1;2 ) = 0 2(c 2 c ) + (1) (d d 2 ) 2 + 2(1) (+2) ( 1;2 2 ) = 0 and they imply 1;2 = 1;2 (2.5.51) c :=c =c 2 (2.5.52) 2 = (1) (d d 2 ) (2.5.53) If we combine the rst equation with (2.5.47), we get 1 is independent of the weight . 1 := 1; = 1;2 = 1; After we plug the expression for d from (2.5.7) we get d = (1) +1 (2a +b + 1 ( + 1)(b b 2 )) Also from the second equation, c seems to be independent of the weight , but with the expression (2.5.38) we conclude that it only depends on the parity of the weight , and moreover a := a is independent of the weight as well. c =a 1 ;odd Remark 2.5.8. Note that, we can not use the coecient of the last term in v @(u ) to impose more relations on the coecients of the dierential. The curl in that term is already equal to zero, since we consider > 0. However, if = 0 then the curl is equal to a right cap and the coecient CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 62 of the last term implies a = 1 This last condition will be conrmed later with more computation on the fantastic ltration, and we note the even weight here as = 0. With imposing the condition, v @(u ) = 0 of the fantastic ltration, denition of the derivation @ is simplied substantially. @ OO +2 ! = 1 OO +2 2 + (b b 2 ) OO +2 N (2.5.54) @ OO OO = 1 OO OO (b b 2 ) OO OO N (2.5.55) @ = a + b 2 N (2.5.56) @ PP =a PP b NN N (2.5.57) @ = (a 1 ;odd ) (2.5.58) + (1) +1 (2a + 1 +b ( + 1)(b b 2 )) N (2.5.59) @ NN = (1) +1 (a 1 ;odd ) NN (2.5.60) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 63 + (1) (2a + 1 +b 2 ( 1)(b 2 b 4 )) NN N (2.5.61) Lemma 2.5.9. For n 1 we have the following identities: @ n = (a + 1 n;odd ) n+1 + (b 2 n(b 2 b 4 ) N (2.5.62) @ NN n = ( 1 n;odd + (1) n++1 (a + 1 ;odd )) NN n+1 (2.5.63) (1) +n (2a + 1 +b 2 ( 1n)(b b 2 )) NN n N (2.5.64) Proof. Identities follow easily using the above denition and the Leibniz rule. After the simplication we get from v @(u ) = 0, we verify that derivation @ respects the odd sl 2 relation (2.4.20) for 0. (< 0 can be shown similarly.) @ 0 B @ OO 1 C A = (a +) X f 1 +f 2 +f 3 =1 (1) n+1 OO +f2 n + (a +) X f 1 +f 2 +f 3 =1 (1) r OO r +f2 (2.5.65) where we use the Leibniz rule and the curl relations (2.4.30). Similarly, below we take the @ of the sum using Leibniz rule along with (2.5.62) and (2.5.63). X f 1 +f 2 +f 3 =1 (1) f2 @ 0 B B @ OO f1 +f2 f3 1 C C A (2.5.66) = X f 1 +f 2 +f 3 =1 0 B B @ ((1) f2 f1;odd + (1) f1+f2++1 (a + ;odd )) OO f1+1 +f2 f3 (2.5.67) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 64 + (1) f1+f2+ (2a +) OO f1 N +f2 f3 + (1) f1++1 (a + f3;odd ) OO f1 +f2 f3+1 1 C C A (2.5.68) An easier way to continue for taking the derivation of this sum is to chop it into two pieces where f 2 is odd and even, since if f 2 is odd then by (2.4.14) and N 2 = 0, the middle term in the sum is 0. One can further divide the sum into smaller pieces depending on the parity of f 1 ,f 3 , and , which make the computation easier. Then it follows that the derivation dened in Denition 2.5.11 respects the odd sl 2 relation (2.4.20) without giving any further relation on the coecients in the denition of @. One can show the same holds for < 0 similarly. We also note that the odd sl 2 relation could be shown before the condition v @(u ) = 0 is veried, however the expression is far more complicated and long, so we preferred to give it after some simplication. Now, we impose the Fantastic ltration condition (2.5.50). v i @(u j ) = X r0 (1) (+j+r+1) 0 B @( 1 r;odd + (1) r+1 (a + 1 ;odd )) (r+i+2)+ (jr1)+ + (1) r+ (2a +b 2 + 1 ( 1r)(b 2 b 4 )) (r+i+1)+ N (jr1)+ + (1) +r+1 (r+i+1)+ @ ( (jr1)+ ) 1 C A Case-I: First, we will study the case 0 i = j 1, note that we already considered i = CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 65 before. Only non-negative degree bubbles line in the sum, therefore the rst term is nonzero if i 2ri 1 and we need to be careful when i = 1 since r must be non-negative. The second and the third term in the sum is nonzero if r =i 1, however derivation of a degree-0 bubble is zero for the third term when r =i 1. For 0i 2 we have: 0 =v i @(u i ) = [( 1 i;even + (1) i (a + 1 ;odd )) + (1) i+1 (2a +b 2 + 1 i(b 2 b 4 )) ( 1 i;odd + (1) i+1 (a + 1 ;odd ))] N Then by studying the all dierent choices of parities for andi, one can deduce that for any and i we have b 2 =i(b 2 b 4 ) which is satised if b :=b 2 =b 4 = 0. This result also simplies d to be d = (1) +1 (2a + 1 ) and gives that d only depends on the parity of the weight . After we update b = 0, then for i = 1 we have 0 =v 1 u 1 = [(1) +1 (a + 1 ;odd ) + (1) (2a + 1 )] N and it gives a = 1 ;even Also, if we combine it with d we get d = 1 If we update := 1 , then the denition of the derivation for the odd nilHecke generators agrees CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 66 with the dierential given in [EQ16c]. @ OO +2 ! = OO +2 2 (2.5.69) @ OO OO = OO OO (2.5.70) Case-II: Now we study the case 0 i < j 1. Then in the expression of v i @(u j ) = 0, the second and third term is nonzero only if j 1 r i 1 which implies j i and it is a contradiction with the assumption. The rst term of the sum is nonzero if j 1ri 2 and it impliesji+1, which is satised only ifj =i+1. Thereforev i @(u j ) = 0 is true forj >i+1 and we consider the case v i @(u i+1 ) = 0 for 0i 2 in detail. 0 =v i @(u i+1 ) = ( i;odd + (1) i+1 ( ;even + ;odd ))1 is even: 1. If i is odd, then v i @(u i+1 ) = 0 follows immediately. 2. If i is even, then v i @(u i+1 ) = 0 implies = 0 The last case makes the derivation trivial and in order to get rid of this problem we change the total order on the index set I =f0; ;g. We declare that for i2I,i + 1<i wheni is even, otherwise we keep the natural order. Example 2.5.10. If = 6 then natural order on I =f0; 1; ; 6g is 0< 1<< 6 and the new total order on I is 1< 0< 3< 2< 5< 4< 6. Note that v i @(u j ) = 0 for ji> 1, and we only need to check v i+1 @(u i ) = 0 for 0i 2 CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 67 and i is even. v i+1 @(u i ) = X r0 (1) (+i+r+1) 0 B @( r;odd + (1) (+r) (r+i+3)+ (ir1)+ + (1) +r+1 (2) (r+i+2)+ N (ir1)+ + (1) +r+1 (r+i+2)+ @ ( (ir1)+ ) 1 C A First term in the sum is nonzero for i 3 r i 1, however we need to be careful for i = 2, since r =i 3< 0 when i = 2. The second and the third terms are nonzero for i 2ri 1. We have, v i+1 @(u i ) = 0 for all even values of i for 0i 2. is odd: 1. If i is even, then v i @(u i+1 ) = 0 follows immediately. 2. If i is odd, then v i @(u i+1 ) = 0 implies = 0. Similar to even case, we solve the triviality problem of derivation implied by the second part above by changing the total order on the index set I. We declare that for i2 I, i + 1 < i when i is odd. Similarly as before, v i @(u j ) = 0 for ji > 1, and we only need to check v i+1 @(u i ) = 0 for 0i 2 and i is odd. Following the similar computation in even case, one can see that v i+1 @(u i ) = 0 follows for all odd values of i for 0i 2. Case-III: As the last case we consider j = and 0i< 1, then 0 =v i @(u ) = ( ;even ;odd + (1) +1 ) (i+1)+ (2.5.71) Note that v i @(u ) = 0 follows immediately unless j6= 1 since negative degree bubbles are zero, CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 68 and for i = 1, we also get v 1 @(u ) = 0. Denition 2.5.11. Derivation on the odd 2-category U satisfying the fantastic ltration (2.5.50) has the following form on the generating 2-morphisms: @ OO +2 ! = OO +2 2 (2.5.72) @ OO OO = OO OO (2.5.73) @ = ;even (2.5.74) @ PP = ;even PP (2.5.75) @ = (1) + N (2.5.76) @ NN = NN NN N (2.5.77) @ OO = ;odd PP (2.5.78) @ OO = 0 (2.5.79) We proved the following: Proposition 2.5.12. Derivation @ given in Denition 2.5.11 is the unique derivation which gives CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 69 rise to a Fantastic ltration on odd 2-category U. Corollary 2.5.13. Derivation of positive degree bubbles are zero. Proof. It follows easily with the Denition 2.5.11. 2.6 Dierentials on the odd 2-category In the previous section, we classied derivations on odd 2-category U which gives rise to a Fantastic ltration and the reason for imposing this ltration is to be able determine the Groethendieck ring K 0 of odd 2-category U after we equip it with dg-structure. In this section, we classify dierentials @ on U which give rise to fantastic ltration. A dierential on the space of 2-morphisms of U is a derivation with @ 2 = 0. By using the Denition 2.5.11 we will take @ 2 of each generator and set it equal to zero in order to determine the denition of the dierential. @ 2 0 B @ OO +2 1 C A = @ 0 B @ OO +2 2 1 C A = 0 (2.6.1) follows easily without giving any further relations on the coecients. @ 2 OO OO = @ OO OO = 0 (2.6.2) is satised as dierential @ of identity morphism is zero. @ 2 = ;even ( ;even +) 2 = 0 (2.6.3) @ 2 PP = ;even ( ;even ) PP 2 = 0 (2.6.4) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 70 Here, for is odd @ 2 of right cup and cap is zero already satised since @ of right cup and cap is zero. If is even then the expression implies the desired result as well. @ 2 = (1) (1) 2 N + (1) +1 2 ! (2.6.5) + (1) N + N N ! = 0 (2.6.6) @ 2 NN = 2 NN N NN N NN N N ! = 0 (2.6.7) @ 2 of left cup and cap is zero follows without giving further relations, and nally we proved the following: Proposition 2.6.1. Derivation @ given in Denition 2.5.11 is the unique dierential on the odd 2-category U which extends the dierential in [EQ16c] and gives rise to a fantastic ltration. 2.7 Quantum sl(2) at a fourth root of unity 2.7.1 Idempotented quantum sl 2 Denition 2.7.1. The quantum group U q (sl 2 ) is the associative algebra (with unit) overQ(q) with generators E, F , K, K 1 and relations 1. KK 1 = 1 =K 1 K, 2. KE =q 2 EK; KF =q 2 FK, 3. EFFE = KK 1 qq 1 . For simplicity the algebra U q (sl 2 ) is written U. CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 71 Dene the quantum integer [a] = q a q a qq 1 with [0] = 1 by convention. The quantum factorial is then [a]! = [a][a 1]::: [1], and the quantum binomial coecient a b = [a]! [b]![ab]! , for 0ba. For a 0 dene the divided powers E (a) = E a [a]! andF (a) = F a [a]! . Denote the Z[q;q 1 ] form of U by A U. This Z[q;q 1 ]-algebra is the Z[q;q 1 ]-subalgebra of U spanned by products of elements in the set n E (a) ; F (a) ; K 1 ja2 Z + o : Denition 2.7.2. The algebra _ U is the non-unital associativeQ(q)-algebra obtained drom U by replacing K and K 1 by a family of orthogonal idempotents 1 for 2 Z, such that (i) 1 1 = ; for all ;2 Z, (ii) E1 = 1 +2 E, F 1 = 1 2 F , (iii) EF 1 FE1 = []1 . Similarly, theA := Z[q;q 1 ]-subalgebra A _ U of _ U is is the Z[q;q 1 ] submodule spanned by products of divided powers E (a) 1 = 1 +2a E (a) ; F (a) 1 = 1 2a F (a) : (2.7.1) There are direct sum decompositions of algebras _ U = M ;2Z 1 _ U1 A _ U = M ;2Z 1 ( A _ U)1 with 1 ( A _ U)1 the Z[q;q 1 ]-subalgebra spanned by 1 E (a) F (b) 1 and 1 F (b) E (a) 1 for a;b2 Z + . Lusztig's canonical basis _ B of _ U consists of the elements (i) E (a) F (b) 1 for a,b2 Z + , n2 Z, ba, (ii) F (b) E (a) 1 for a,b2 Z + , 2 Z, ba, where E (a) F (b) 1 ba = F (b) E (a) 1 ba . The importance of this basis is that the structure constants CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 72 are in N[q;q 1 ]. In particular, for x;y2 _ B xy = X x2 _ B m z x;y z with z2 _ B and m z x;y 2 N[q;q 1 ]. Let m _ B n denote the set of elements in _ B belonging to 1 _ U1 . Then the set _ B is a union _ B = a ;2Z m _ B n : 2.7.2 Small quantum sl 2 The small quantum group introduced by Lusztig is a nite dimensional Hopf algebra over the eld of cyclotomic integers [Lus90]. Here we consider the small quantum group at a fourth root of unity. Let p 1 be a primitive fourth root of unity and consider the ring of cyclotomic integers Z[ p 1] = Z[q;q 1 ]= 4 (q) = Z[q;q 1 ]=(1 +q 2 ); (2.7.2) where n denote the nth cyclotomic polynomial. Denote by _ U Z[ p 1] the idempotented Z[ p 1]- algebra dened by change of basis _ U Z[ p 1] = A _ U Z[q;q 1 ] Z[ p 1]: Set [k] p 1 to be the quantum integer [k] evaluated at p 1. The divided power relation implies that in _ U Z[ p 1] the elements E k 1 = [k] p 1 E (k) 1 ; F k 1 = [k] p 1 F (k) 1 (2.7.3) are only nonzero when 0k 2. Denition 2.7.3. The small quantum group _ u p 1 (sl 2 ) is the non-unital associative algebra given by the Z p 1 -subalgebra in _ U p 1 generated by the elements fE1 ; F 1 ; 1 ; 2 Zg: CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 73 It has a presentation over Z[ p 1] given by (i) 1 1 = ; (ii) E1 = 1 +2 E; F 1 = 1 2 F , (iii) EF 1 FE1 = [] p 1 1 , (iv) E 2 = 0; F 2 = 0. The algebra _ u p 1 (sl 2 ) inherits a Z[ p 1]-integral basis by reduction of Lusztig's canonical basis. We denote byB( p 1) the following elements B( p 1) := n E (a) F (b) 1 ja;b2f0; 1g; ba o [ n F (b) 1 E (a) ja;b2f0; 1g; ba o (2.7.4) with it understood that E (1) F (1) 1 ba =F (1) E (1) 1 0 . 2.8 Categorication results 2.8.1 Divided power modules In [EKL14] it was shown that ONH n has a unique graded indecomposable projective moduleP n and that there is an algebra isomorphism ONH n = Mat On (P n ); (2.8.1) where O n is the superalgebra of odd symmetric polynomials. In [EQ16c] they equip P n with a dg- module structure compatible with the dierential on ONH n and denote the resulting (OPol n ; O n )- bimodule by Z n . Theorem 2.8.1. 1. There is an equivalence of dg algebras (ONH n ;@)! END O op n (Z n ): (2.8.2) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 74 (Corollary 3.9 [EQ16c]). 2. For any n 0, Z n is a nite-cell right dg-module over O n ([EQ16c, Proposition 3.16]). 3. If n 2, then ONH n is an acyclic dg-algebra. Consequently, the derived categoryD(ONH n ) is equivalent to the zero category ([EQ16c] Proposition 3.16). 4. As a left ONH n dg module, Z n is only cobrant if n = 0; 1 and is acyclic otherwise [EQ16c, Proposition 3.17]. In light of the above theorem, we denote the dg-module Z n by E (n) + as (2.8.2) gives a dg- categorication of the divided power relation E n = [n]!E (n) . 2.8.2 The DG-Grothendieck ring In this section, we state the main theorems on the DG-Grothendieck ring of the odd 2-category U. The corresponding proofs and more detailed discussion of these theorems can be found in [EL18]. Denote the abelian category of DG-modules over (U;@) by U @ mod. It decomposes into a direct sum of dg-categories U @ mod = M ; ( U ) @ mod: (2.8.3) Composition of 1-morphisms induces induction functors ( 4 U 3 2 U 1 ) @ mod! 2;3 ( 4 U 1 ) @ mod (2.8.4) MN7! Ind(MN ) for any 1 ; 2 ; 3 ; 4 2 Z. At the level of derived categories, the induction functor gives rise to an exact functor Ind:D(U U;@)!D(U;@) (2.8.5) and Z[ p 1]-linear maps [Ind]: K 0 (D(U U;@))!K 0 (U;@): (2.8.6) CHAPTER 2. A DG-STRUCTURE ON ODD 2-CATEGORY 75 For xed 2 Z dene U := M 2Z U : (2.8.7) Proposition 2.8.2. There is a derived equivalence D(U ) =D(END U (X + )) (2.8.8) Corollary 2.8.3. For any weight2 Z, the Grothendieck group of the DG-category U is isomorphic to the corresponding span of canonical basis elements K 0 (U ) = Z[ p 1]h _ B( p 1) i: As a consequence of strong positivity we also have the following result. Corollary 2.8.4. For any weights 1 ; 2 ; 3 ; 4 2 Z, the DG-categories 4 U 3 , and 2 U 1 have the Kunneth property K (4 U 3 ) Z[ p 1] 2 U 1 =K 0 ( 4 U 3 2 U 1 ): Theorem 2.8.5. There is an isomorphism of Z[ p 1]-algebras _ u Z[ p 1] (sl(2))!K 0 (U;@) (2.8.9) that sends E1 7! [E1 ] and 1 F7! [1 F] for any weights 2 Z. Chapter 3 A DG-extension of Symmetric Functions 3.1 The nilHecke algebra Many of our constructions for the extended nilHecke algebra build o of results for the usual nilHecke algebra and its action on polynomials. Here we recall the relevant results. 3.1.1 The denition Recall the nilHecke algebra NH n dened by generators x i for 1 i n and @ j for 1 j n 1 and relations x i x j =x j x i ; @ i x j =x j @ i ifjijj> 1; @ i @ j =@ j @ i ifjijj> 1; @ 2 i = 0; @ i @ i+1 @ i =@ i+1 @ i @ i+1 ; x i @ i @ i x i+1 = 1; @ i x i x i+1 @ i = 1: (3.1.1) It is not hard to prove that these relations imply @ i x a+1 i x a+1 i+1 @ i = h a (x i ;x i+1 ) =x a+1 i @ i @ i x a+1 i+1 : (3.1.2) A version of the work presented in this chapter is available in Journal of Combinatorial Algebra [AEHL17]. 76 CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 77 for all a> 0. Given any elementw2 S n and a reduced decompositionw =s i1 :::s im into simple transpositions we write @ w := @ i1 :::@ im . The axioms ensure this denition does not depend on the choice of reduced expression. We write w 0 for the longest word in the symmetric group S n and @ w0 for the corresponding product of divided dierence operators. The algebraNH n acts on the polynomial ringP n :=Q[x 1 ;:::;x n ] withx i acting by multiplication by x i and @ i : P n ! P n given by divided dierence operators @ i := 1s i x i x i+1 : (3.1.3) We recall several important facts relating to the nilHecke algebra and its action on polynomials. The ring of symmetric functions can be realized strictly in terms of the divided dierence operators n := Z[x 1 ;:::;x n ] Sn = n1 \ j=1 ker@ i = n1 \ j=1 im@ i : The additive basis of n given by Schur functions s can be dened using the nilHecke algebra action on polynomials via s :=@ w0 (x + ) :=@ w0 (x n1+1 1 x n2+2 2 :::x 0+n n ); for = ( 1 ;:::; n ) a partition with n parts. For w2 S n dene the Schubert polynomials of Lascoux and Sch utzenberger [LS82] as S w (x) =@ w 1 w0 x (3.1.4) where w 0 is the permutation of maximal length and x =x a1 1 x a2 2 x a1 . In case w = 12 S n , we have S id =@ w0 (x ) = 1. We have im@ w0 = n P n : (3.1.5) CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 78 Indeed, if f 2 n , then f = f@ w0 (x ) = @ w0 (fx ) since divided dierence operators are n -linear. Conversely, if f2 im@ w0 , then @ i (f) = 0 for i = 1;:::;n 1, hence f2 P Sn n . The polynomial ring P n is a free module over n of rank n! [Man01, Proposition 2.5.5 and 2.5.5]. In particular, multiplication in P n induces a ring isomorphism P n 'H n n where H n is equivalently the abelian subgroup spanned by either of the sets fS w jw2 S n g or x i1 1 :::x in n j 0i k nk: . The last statement allows us to identify End n (P n ) as the matrix ring of size n! with coecients in the ring n . The ring P n is graded with deg(x i ) = 2. Taking grading into account, it follows that there is an isomorphism of graded rings End n (P n ) = Mat((n) ! q 2 ; n ), where (n) ! q 2 = q n(n1)=2 [n]! are the symmetric quantum factorials [Lau08, Proposition 3.5]. The action of NH n on P n denes a graded ring homomorphism : NH n ! Mat((n) ! q 2; n ): It was shown in [Lau08, Proposition 3.5] that is an isomorphism of graded rings. We recall an alternative proof from [KLMS12] Section 2.5 that we translate into algebraic language from the so-called thick calculus. For any composition = ( 1 ;:::; n ) writex :=x 1 1 x 2 2 :::x n n . We writex :=x n1 1 x n2 2 :::x 0 n . The set of sequences Sq(n) :=f` =` 1 :::` n1 j 0` ; = 1; 2;:::n 1g (3.1.6) has sizejSq(n)j =n!. Letj`j = P ` , and set b ` j =j` j . Dene a composition with n-parts by b ` = (0; b ` 1 ;:::; b ` n1 ) = (0; 1` 1 ; 2` 2 ; ;n 1` n1 ): (3.1.7) Lete (a) r denote therth elementary symmetric polynomial ina variables. The standard elementary monomials are given by e ` := e (1) `1 e (2) `2 :::e (a1) `a1 : (3.1.8) CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 79 Dene elements in NH n by ` := e ` @ w0 ; ` := (1) b ` x @ w0 x b ` : (3.1.9) Theorem 3.1.1 ([KLMS12]). 1. For all `;` 0 in Sq(n), ` 0 ` = `;` 0 x @ w0 . 2. The set ` ` 2 Sq(n) form a complete set of mutually orthogonal primitive idempotents in NH n . 3. The identity element 12 NH n decomposes as 1 = X ` (1) b ` e ` @ w0 x b ` : (3.1.10) 4. Enumerate the rows and columns of n!n!-matrices by the elements `2 Sq(n). There is an isomorphism of graded algebras Mat (n) ! q 2; n ! NH n (3.1.11) sending an element x2 ext n in the (`;` 0 ) entry to the element ` x ` 0. The nilHecke algebra is the simplest example of a KLR-algebra, corresponding to the Lie algebra sl 2 . The results above are critical in the categorication of positive parts of quantized universal enveloping algebras via KLR-algebras [KL09, KL11, Rou08]. Another important construction from categoried representation theory is the so-called cyclotomic quotients of KLR-algebras. These are used to categorify irreducible representations of U q (g). For each N > 1 dene the cyclotomic ideal of NH n as the two sided ideal generated by x N 1 , I N :=hx N 1 i: (3.1.12) We dene the cyclotomic quotient by NH N n := NH n =I N . We have the following results. CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 80 The isomorphism from (3.1.11) induces an isomorphism [Lau12, Proposition 5.3] Mat (n) ! q 2;H (Gr(n;N)) ! NH N n (3.1.13) whereH (Gr(n;N)) is the cohomology ring of the Grassmannian of complex n-planes inC N . The categories of graded projective modules over L n NH N n categorify [LV11, KK11, Web13] the irreducible U q (sl 2 ) representation V N of highest weight N. 3.2 The extended nilHecke algebra 3.2.1 The denition The extended nilHecke algebra NH ext n , rst dened in [NV16], is a bigraded algebra with genera- tors x 1 ; ;x n , @ 1 ; ;@ n1 , generators ! 1 ; ;! n satisfying equations (3.1.1) and the following relations x i ! j =! j x i ; ! i ! j =! j ! i ; @ i ! j =! j @ i ij ! i+1 (x i+1 @ i @ i x i+1 ): For each xed integer k the algebra NH ext n admits a Z Z-grading in which the generators x i ;@ i ;! i are bihomogeneous with degrees deg(x i ) = (2; 0); deg(@ i ) = (2; 0); deg(! k ) = (2k; 1): (3.2.1) If a2 NH ext n is homogeneous with deg(a) = (i;j), then i =: deg q (a) is referred to as the quantum degree and j =: deg h (a) is the homological degree. The parity of a is by denition the homological degree modulo 2. Remark 3.2.1. For each m2 Z we may put a bigrading on NH ext n by leaving deg(x i ) and deg(@ i ) unchanged, while shifting the degrees of ! i by declaring deg(! k ) = (2(k +m); 1). The relations are homogeneous with respect to this bigrading, regardless of m. The resulting bigraded rings will be denoted (NH ext n ) (m) . Note that the algebra (NH ext n ) (m) is naturally a graded subalgebra of NH ext m+n CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 81 given by restricting to the generators fx i ;@ j ;! i jm + 1in +m; m + 1j <n +m 1:g Remark 3.2.2. In [NV16] they consider an additional grading for their application to categorical Verma modules. Here we ignore this additional grading. 3.2.2 Action on polynomials Dene the extended polynomial ring P ext n =Q[x 1 ;:::;x n ] ^ [! 1 ;:::;! n ]; (3.2.2) bigraded via deg(x i ) = (2; 0), deg(! i ) = (2i; 1). Then P ext n has the structure of a bigraded NH ext n - module, dened by lettingx i and! i act by left multiplication and letting@ i act by extended divided dierence operators @ i (1) = 0; @ i (! j ) = ij ! j+1 ; @ i (x j ) = 8 > > > > > > < > > > > > > : 1 if j =i; 1 if j =i + 1; 0 otherwise. These operators are extended to arbitrary polynomials by the rule @ i (fg) =@ i (f)g +f@ i (g) (x i x i+1 )@ i (f)@ i (g) (3.2.3) for all f;g2Q[x 1 ;:::;x n ] V [! 1 ;:::;! n ]. CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 82 3.2.3 Dierentials Recall that a dierential graded algebra (or DG-algebra) is a Z-graded unital algebraA withd: A! A which is degree -1 satisfying d 2 = 0; d(ab) =d(a)b + (1) deg(a) ad(b); d(1) = 0: (3.2.4) A left DG-module M is a graded left A-module with dierential d M : M i !M i1 such that for all a2A, m2M, d M (am) =d(a)m + (1) deg(a) ad M (m): (3.2.5) Remark 3.2.3. In the discussion below, we will consider bigraded algebras and modules with dierentials. Despite the presence of two gradings, we will continue to use the standard abbreviation and refer to them simply as DG algebras and modules. For each N > 0, dene a dierential d N on NH ext n of bidegree (2N + 2;1) by d N (x i ) = 0; d N (@ i ) = 0; d N (! i ) = (1) i h Ni+1 (x i ); (3.2.6) where x i denotes the set of variablesfx 1 ;x 2 ;:::;x i g. Note the ordinary nilHecke algebra NH n is in the kernel of this dierential for all N. Furthermore, d N (! 1 ) =x N 1 . By [HL10, Proposition 2.8] it follows d N (! j ) is contained in the cyclotomic ideal I N :=hx N 1 i from (3.1.12). Theorem 3.2.4 ([NV16] Proposition 8.3). The DG-algebra NH ext n ;d N is quasi-isomorphic to the cyclotomic quotient of the nilHecke algebra NH N n := NH n =I N . CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 83 3.3 The ring of extended symmetric polynomials 3.3.1 Denition Preliminary denition The action of NH ext n on the extended polynomial ring P ext n =Q[x 1 ;:::;x n ] V [! 1 ;:::;! n ] gives rise to a homomorphism NH ext n ! End Q P ext n : By analogy with the case of symmetric polynomials, we dene the ring of extended symmetric polynomials ext n as ext n = n1 \ i=1 ker@ i = n1 \ i=1 im@ i : Remark 3.3.1. The ring ext n P ext n is bigraded and graded commutative (that is, super-commutative) with respect to the parity discussed in the comments following (3.2.1). Action of the symmetric group on P ext n The standard action of the symmetric group S n on the polynomial ring P n =Q[x 1 ;:::;x n ] lifts to an action on P ext n . Namely, one sets s i (x j ) =x si(j) and s i (! j ) =! j + ij (x j x j+1 )! j+1 (3.3.1) for any 1in 1, 16=jn, and extends it to P ext n by s i (fg) =s i (f)s i (g) for any f;g2 P ext n . With respect to this action, the operators@ i coincide with the standard divided dierence operators: @ i = ids i x i x i+1 : (3.3.2) In particular, (3.2.3) reduces to the standard Leibniz rule for divided dierence operators @ i (fg) =@ i (f)g +s i (f)@ i (g): (3.3.3) It follows that ext n coincides with the subalgebra of S n {invariants ext n = (P ext n ) Sn . CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 84 We now provide an explicit description of ext 2 and ext 3 . The general case is discussed in 3.3.2 and 3.3.4. Remark 3.3.2. The algebra P ext n is endowed with another, more natural action of the symmetric group (which on the other hand does not respect theZ{grading (3.2.1) and does not extend to an action of NH ext n ). Namely, for anyw2 S n , one can setw(! i ) =! w(i) . The corresponding subalgebra of S n {invariants is described by Solomon in [Sol63], see also [Kan01, Chapter 22]. In Section 3.4, we discuss the connection between these two actions and their invariants. Case n = 2 The algebra P ext 2 is a free module of rank 4 over P 2 , and it is easy to see that ext 2 is a free module of rank 4 over 2 with basisf1;! 1 +A! 2 ;! 2 ;! 1 ! 2 g, where A is any solution of @ 1 (A) = 1. Particular choices of A arefx 1 ;x 2 ; 1 2 (x 1 x 2 )g. Case n = 3 The algebra P ext 3 is a free module of rank 8 over P 3 . Then v =a +b! 1 +c! 2 +d! 3 +e! 1 ! 2 +f! 1 ! 3 +g! 2 ! 3 +h! 1 ! 2 ! 3 2 ext 3 if and only if a2 3 , b =@ 1 @ 2 (d), c =@ 2 (d), e =@ 2 @ 1 (g), f =@ 1 (g), h2 3 , and @ 1 (d) = 0 @ 1 @ 2 (d)2 3 @ 2 (g) = 0 @ 2 @ 1 (g)2 3 : It is easy to show that the general solution of the system @ 1 (d) = 0;@ 1 @ 2 (d)2 3 has the form d =Af 1 +Bf 2 +f 3 ; where f 1 ;f 2 ;f 3 2 3 and A;B2 P 3 are any solution of @ 1 (A) = 0; @ 1 (B) = 0; @ 1 @ 2 (A) = 1; @ 2 (B) = 1: CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 85 Similarly for g. We conclude that ext 3 is a free module over 3 of rank 8 with basisf1; ! 1 + @ 2 (A)! 2 +A! 3 ; ! 2 +B! 3 ; ! 3 ; ! 1 ! 2 +@ 1 (C)! 1 ! 3 +C! 2 ! 3 ; ! 1 ! 3 +D! 2 ! 3 ; ! 2 ! 3 ; ! 1 ! 2 ! 3 g where A;B;C;D2 P 3 are any solution of @ 1 (A) = 0 @ 1 (B) = 0 @ 2 (C) = 0 @ 1 (D) = 1 @ 1 @ 2 (A) = 1 @ 2 (B) = 1 @ 2 @ 1 (C) = 1 @ 2 (D) = 0 Particular choices of solutions of the above system are A2fx 1 x 2 ;x 2 3 g,B2fx 1 +x 2 ;x 3 g,C =x 2 1 , and D =x 1 . 3.3.2 The size of extended symmetric functions We now discuss the general case for n 3. Notations For any binary sequence 2Z n 2 , set ! =! 1 1 ! n n . Then P ext n = M 2Z n 2 P n ! : The action of S n is concisely described by the formula s i (! ) =! + i;1 i+1;0 (x i x i+1 )! si() : For k = 1;:::;n 1 and 2Z n 2 , set I k =f2Z n 2 j k = 1; k+1 = 0g; J k =f2Z n 2 j k = 0; k+1 = 1g =s k (I k ); D =fkj2J k g; CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 86 so that, in particular, s k (! ) = 8 > < > : ! if 62I k ; ! + (x i x i+1 )! si() if 2I k : For k = 0; 1;:::;n, let (Z n 2 ) k be the subset of strings of length k (Z n 2 ) k =f2Z n 2 jjj = n X i=1 i =kg endowed with the following partial ordering. We say that if there exists a sequence in (Z n 2 ) k 1 =; 2 ;:::; m = where m > 1 and for any i = 1;:::;m 1, i 2 I r and i+1 2 J r for some r. Let (k) ; (k) be, respectively, the highest and lowest element in ((Z n 2 ) k ;),i.e. (k) i = 0 if and only if i<nk + 1 and (k) i = 0 if and only if i>k. Grassmannian permutations A Grassmannian permutationw is a permutation with a unique descent. In other words there exists k2f1;:::;n 1g such that w(i)<w(i + 1) if i6=k and w(k)>w(k + 1). The Grassmannian permutations with descent nk are in canonical bijection with elements in (Z n 2 ) k , as we now describe. Let 2 (Z n 2 ) k be given. Let 1 v 1 < < v nk n be the indices such that v1 = = v nk = 0, and let 1 u 1 < < u k n be the indices such that u1 = = u k = 1. Dene 2 S n by (i) = 8 > < > : v i if 1ink u in+k if nk + 1in CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 87 More concisely, is the unique minimal length permutation which sends (k) = ( 0;:::; 0 | {z } nk ; 1;:::; 1 | {z } k )7!: In particular, (k) = id. Note that is a minimal length representative of a coset in S n =S nk S k . For every 2 (Z n 2 ) k , 6= (k) , has a unique descent at nk, and it is therefore Grassman- nian. Conversely every Grassmannian permutation arises in this way. Lehmer codes and partitions Recall that the Lehmer code of a permutation w is the composition L w = (L w 1 ;:::;L w n ), where L w i = #fi<j : w(j)<w(i)g: We write (w) for the partition obtained by sorting L w into decreasing order. In particular, the Lehmer code of the Grassmannian permutation , 2 (Z n 2 ) k , is given by L i = 8 > < > : m if u m m<i u m+1 (m + 1); 0 if u m+1 (m + 1)<i: More concretely, if 1 i nk, L i is the number of ones which appear to the left of the i-th zero of , and L i = 0 otherwise. In particular, L w 1 L w nk , and L w i = 0 for i>nk. The partition corresponding to is then :=( ) = (m rm ) m=k;:::;1 : where r m = u m+1 u m 1 for every m = 0;:::;k (we impose u 0 = 0;u k+1 = n + 1). Notice that has at mostnk non zero terms. In fact, one sees immediately that the biggest possible size of CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 88 the tableau of shape , 2 (Z n 2 ) k , is (nk)k. The conjugate partition is 0 = ( 0 j ) j=1;:::;k 0 j = k X m=j r m =n + 1u j (kj + 1) =nku j +j: Examples For any 1j <kn, setc [j;k] =s j s k1 andc (k) =c [k;n] c [2;nk+2] c [1;nk+1] . We sometimes write c [j] :=c [j;n] . It may be helpful to visualize these elements c (k) = 1 nk nk+1 n where diagrams are read from bottom to top. Then it is easy to see that c (k) ( (k) ) = (k) and, for any 2 (Z n 2 ) k , is a subword of c (k) . Main result The rest of this section is devoted to prove the following Theorem 3.3.3. (i) The ring of extended symmetric polynomials ext n is a free module over n of rank 2 n . (ii) For any collection of polynomialsfp g 2Z n 2 satisfying p 2 P S njj S jj n and @ p = 1 (3.3.4) there is an isomorphism of n {modules ext n ' M 2Z n 2 n ! s (p ) where ! s (p ) :=! + X @ (p )! CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 89 (iii) Multiplication in ext n induces a ring isomorphism ext n ' n ^ [! s 1 ;:::;! s n ]: (iv) Multiplication in P ext n induces a ring isomorphism P ext n 'H n ext n , whereH n P n is the subspace spanned by either of the setsfS w jw2 S n g or x i1 1 :::x in n j 0i k nk: . This gives rise to a canonical ring isomorphism End ext n (P ext n )' Mat(n!; ext n ): Remark 3.3.4. In 3.3.4 we construct examples of p 2 P S njj S jj n satisfying (3.3.4), for each . The proof is carried out in 3.3.2{3.3.2. First characterization of ext n Proposition 3.3.5. Let v = P f ! 2 P ext n , with f 2 P n . The following are equivalent. (i) v2 ext n (ii) For every i = 1;:::;n 1, @ i (f ) = 8 > < > : 0 if 62J i ; f si() if 2J i : (3.3.5) (iii) For every 2Z n 2 , @ i (f ) = 8 > < > : 0 if i62D ; f si() if i2D : (3.3.6) Proof. Clearly, (ii) and (iii) are equivalent. Now, let v = P f ! , f 2 P n . For every i = 1;:::;n 1, s i (v) = X 2Z n 2 s i (f ) + X 2Ii (x i x i+1 )s i (f )! si() = CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 90 = X 62Ji s i (f )! + X 2Ji s i (f ) +s i (f si() )(x i x i+1 ) ! Therefore v2 ext n if and only if, for every i = 1;:::;n 1, @ i (f ) = 8 > < > : 0 if 62J i ; s i (f si() ) if 2J i : Finally, one observes that for every 2 J i , s i ()62 J i . Therefore, s i (f si() ) = f si() and (i) is equivalent to (ii). Simplication The system of equations (3.3.6) preservesjj, i.e. there are n + 1 independent sets of equations, for k = 0; 1;:::;n, 82 (Z n 2 ) k @ i (f ) = 8 > < > : 0 if i62D ; f si() if i2D Let (k) ; (k) be, as before, the highest and lowest element in (Z n 2 ) k with respect to. Then it follows from (3.3.6) that f (k)2 n and, for every 2 (Z n 2 ) k , f =@ (f (k)) In particular, any solution of (3.3.6) is determined by the elements f (k)2 P n ,k = 0; 1;:::;n. More specically, we have the following Corollary 3.3.6. Let v = P f ! 2 P ext n with 2Z n 2 and f 2 P n . Then v2 ext n if and only if, for any k = 0;:::;n 1, the elements F k :=f (k) satisfy (i) F k 2 P S nk S k n ; (ii) for every 2 (Z n 2 ) k , f =@ (F k ). Proof. Note that if = (k) , then D =fnkg. Thus, the necessity of conditions (i) and (ii) are easy consequences of condition (iii) of Proposition 3.3.5. CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 91 Now we show that (i) and (ii) are sucient conditions for membership v 2 ext n . Fix k 2 f1;:::;ng, and suppose F k 2 P n is given and satises (i). Dene f :=@ (F k ) for all 2 (Z n 2 ) k , and setv := P 2(Z n 2 ) k f ! . We must show that@ i (f ) = 0 wheneveri62D . Letw be the longest element of S nk S k S n . By (i), F k 2 P S nk S k n is symmetric in the rst nk variables and the lastk variables. It follows thatF k =@ w (G k ) for some polynomialG k . This is a straightforward generalization of the fact that n = im@ w0 P n , and follows easily from properties of the nilHecke algebra. From the denition of D , it is clear that `(s i w) =`( w) 1 if and only if i62D . Recall that, for any ; 0 2 S n , @ @ 0 = 8 > < > : @ 0 if `( 0 ) =`() +`( 0 ) 0 otherwise (see, for example, [Man01,x2.3.1]). Thus, ifi62D ,@ i (f ) =@ i @ @ w (G k ) = 0. This completes the proof. Proof of Theorem 3.3.3 Corollary 3.3.6 gives us a map of n -modules k : P S nk S k n ! ext n dened by k (F ) := X 2(Z n 2 ) k @ (F )! : Clearly k is injective, since F can be recovered as the coecient of ! (k) in k (F ). By Corollary 3.3.6, k surjects onto the component of ext n consisting of elements which are degreek in the exterior variables ! i . Since the dimension of P S nk S k n over n = P Sn n is n k , statement (i) of Theorem 3.3.3 follows. Now, letfp g 2Z n 2 be a solution of (3.3.4) and set ! s (p ) =! + X @ (p )! By 3.3.6, the elements ! s (p ) belong to ext n and they are linearly independent, since they are CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 92 triangular with respect tof! g. By a dimension argument this induces an isomorphism of n { modules ext n ' M 2Z n 2 n ! s (p ) This proves Theorem 3.3.3 (ii). For (iii), suppose we have chosen elements ! s i =! i + X j>i f j ! j 2 ext n for i2f1;:::;ng. Since these elements are degree 1 in the exterior variables, we have ! s i ! s j =! s j ! s i for every 1 i;j n. Given the triangularity off! s i g with respect tof! i g, the resulting map of rings ^ [! s 1 ;:::;! s n ]! ext n (3.3.7) is clearly injective. Extending linearly in n gives an injective map of n -algebras : n ^ [! s 1 ;:::;! s n ]! ext n : By a dimension argument, is surjective, and we obtain (iii). Finally, extending (3.3.7) by P n {linearity gives a P n {algebra homomorphism P n ^ [! s 1 ;:::;! s n ]! P n ^ [! 1 ;:::;! n ] = P ext n ; (3.3.8) which we claim is an isomorphism. Namely, the homomorphism is induced by the nilpotent matrix A with coecients in P n such that ! s = (I +A)! () ! = n1 X i=0 (1) i A i ! s where ! and ! s denote, respectively, the column vectors (! 1 ;:::;! n ) T and (! s 1 ;:::;! s n ) T . This CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 93 determines the inverse to (3.3.8). Applying the classical identication P n 'H n n , we get a ring isomorphism P ext n 'H n n ^ [! s 1 ;:::;! s n ]'H n ext n : In particular, P ext n is a free module of rank n! over ext n and there is a canonical isomorphism End ext n (P ext n )' Mat(n!; ext n ) which completes the proof of Theorem (3.3.3). 3.3.3 Structure of the extended nilHecke algebra The above analysis of ext n also has consequences for NH ext n . Recall that c[j] = c[j;n] denotes the permutation s j s n1 . Recall also that that ext n is bigraded b Proposition 3.3.7. Let p j 2 P Sn1S1 n be polynomials of degree nj such that @ c[j] (p j ) = 1, and set ! s j := P i @ c[i] (p j )! i as in Theorem 3.3.3. Then there is an isomorphism of algebras NH ext n = NH n Q ^ [! s 1 ;:::;! s n ]: The induced action on P ext n = P n Q V [! s 1 ;:::;! s n ] is the standard action of NH n on P n , tensored with the exterior algebra. Consequently, NH ext n = End ext n (P ext n ) = Mat((n) ! q 2; ext n ) (3.3.9) where ext n acts on P ext n by right multiplication. Note that ext n is graded commutative, hence in order for left multiplication by ! i 2 NH n on P ext n to honestly commute with the action of ! s j 2 ext n (as opposed to commutativity up to sign), it is necessary to let ext n act on P ext n by right multiplication in (3.3.9). Proof. By denition NH ext n containsNH n andP ext n as subalgebras. Tensoring the inclusion maps gives us an algebra map NH n Pn P ext n ! NH ext n : CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 94 By Theorem 3.3.3, we know that P ext n = P n Q V [! s 1 ;:::;! s n ], hence the above reduces to an algebra map NH n Q ^ [! s 1 ;:::;! s n ]! NH ext n : As a NH n -module, the right hand side is isomorphic to NH n Q V [! 1 ;:::;! n ]. From the denitions, it is clear that the ! s i are unitriangular with respect to the ! i , hence the above algebra map is an isomorphism. This proves the rst statement. The statement regarding the action on P ext n is easily veried. Finally (3.3.9) follows by combining the standard fact that NH n = End n (P n ) together with Theorem 3.3.3, which states that P ext n is free of rank [n]! over ext n . As an immediate corollary we have the following analogue of the usual fact that n =Z(NH n ). Corollary 3.3.8. ext n is isomorphic to the graded center of NH ext n as graded algebras. Here, the graded center of a Z=2 graded algebra A = A 0 A 1 is spanned by homogeneous elements z2 A such that za = (1) deg(a) deg(z) az for every homogeneous a2 A. Here the Z=2 grading on NH ext n is inherited from the homological grading as in the comments following (3.2.1). 3.3.4 Bases of ext n We now discuss some explicit examples of bases of ext n . We adopt the following criteria. From Theorem 3.3.3, a basis of ext n is determined by any family of elementsfp j g 1jn P n satisfying @ c[j] (p j ) = 1 and p j 2 P Sn1S1 n (3.3.10) This allows to construct a ring isomorphism ext n ' n ^ [! s 1 ;:::;! s n ] where ! s j = X kj @ c[k] (p j )! k : Any such collectionf! s j g 1jn will be referred to as an exterior basis of ext n . CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 95 Schubert polynomials The rst example we discuss involves the use of Schubert polynomials. Recall that the Schubert polynomials S w 2 P n , with w2 S n , are a collection of polynomials indexed by elements of S n and characterized by the following conditions: (i) S id = 1; (ii) for every u2 S n @ u S w = 8 > < > : S wu 1 if l(wu 1 ) =l(w)l(u) 0 otherwise. More explicitly, one can check that S w =@ w 1 w0 x n1 1 x n2 2 x n1 : Schubert polynomials and ext n The above characterization implies immediately the following Proposition 3.3.9. The elements p j = S c[j] , 1jn, are a solution of (3.3.10). In particular, the elements # j =! j + X k>j S c[j;k] ! k dene an exterior basis of ext n . Proof. It is clear from the denitions that S c[j] satisfy (3.3.10). The proposition follows by an application of Theorem 3.3.3. It is interesting to observe that the Schubert polynomials allow to dene a solution to the full system (3.3.4). Specically, for every 2 (Z n 2 ) k , we can set p = S . Then, it is easy to see that @ S = S 1 , @ S = 1 and @ i S = 0 CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 96 for every i6=nk. Therefore, we get extended symmetric polynomials # =! + X S 1 ! 2 ext n : In fact, these are exactly the elements of the standard basis of V [# 1 ;:::;# n ]. Proposition 3.3.10. The standard basis of V [# 1 ;:::;# n ] has the following description. For any 2 (Z n 2 ) k , # 1 1 # n n =! + X S 1 ! The proof will be carried out in 3.3.4, 3.3.4 and 3.3.4. Determinantal identities In what follows we will make use of the following result, relating Schubert polynomials of Grass- mannian permutations to Schur functions. Proposition 3.3.11 (Proposition 2.6.8[Man01]). If w2 S n is a Grassmannian permutation, and if r is its unique descent, then S w = s (w) (x 1 ;x 2 ; ;x r ); where s (w) is the Schur function in the variablesfx 1 ;:::;x r g corresponding to the partition (w). Example 3.3.12. 1. If w = c[j] = s j s j+1 :::s n1 2 S n , then w has a unique descent at position n 1. The Lehmer code is (0;:::; 0; 1;:::; 1; 0) and the corresponding partition (w) = (1 nj ). Hence, S c[j] = e nj (x 1 ;:::;x n1 ). 2. More generally, if j < k and w = c[j;k] = s j s j+1 :::s k1 2 S n , then w is a Grassmannian permutation with a unique descent in positionk1. The corresponding partition(w) = (1 kj ) and S c[j;k] = e kj (x 1 ;:::;x k1 ). 3. The permutations c (k) = c[k;n]c[2;nk + 2]c[1;nk + 1] have a unique descent at position nk. The Lehmer code for c (k) has L w 1 = L w 2 = = L w nk = k and L w j = 0 for j >nk. It follows that S c (k) = s (k nk ) (x 1 ;x 2 ;:::;x nk ). CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 97 4. Generalizing all of the previous examples, w =c[k;n]c[k 1;n 1]c[kj + 1;nj + 1] has a unique descent at nj, and S w = s (j nk ) (x 1 ;x 2 ;:::;x nj ). Recall that Schur functions satisfy the second Jacobi{Trudi identity: for every partition of length l() s = det(e 0 i +ji ) l( 0 ) i;j=1 = det(e 0 j +ij ) l( 0 ) i;j=1 (3.3.11) where 0 is conjugate to . The proof of Proposition 3.3.10 relies on the following Lemma 3.3.13. For any 2 (Z n 2 ) k , let u = (u 1 ;:::;u k ) and be, respectively, the corresponding sequence of indices and the partition dened in 3.3.2. Then s = det(e (nk+i)uj ) k i;j=1 Moreover, s (x 1 ;:::;x nk ) = det(e nk+iuj (x 1 ;:::;x nk )) k i;j=1 = det(e nk+iuj (x 1 ;:::;x nk+i1 )) k i;j=1 Example 3.3.14. The result of Lemma 3.3.13 is addressing the following phenomenon. Set n = 2 and consider the permutation s 2 s 1 . In this case we get S s2s1 =x 2 1 = det 2 6 4 x 1 1 x 1 x 2 x 1 +x 2 3 7 5 = det 2 6 4 e 1 (x 1 ) 1 e 2 (x 1 ;x 2 ) e 1 (x 1 ;x 2 ) 3 7 5 On the other hand, s 2 s 1 has a unique descent at 1, its partition is [2], its conjugate is [1; 1], and, by the second Jacobi{Trudi identity, s [2] = det 2 6 4 e 1 1 e 2 e 1 3 7 5 = e 2 1 e 2 These coincide when we input the set of variablesfx 1 g, namely S s2s1 =x 2 1 = e 2 1 (x 1 )e 2 (x 1 ) = s [2] (x 1 ): CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 98 Proof of Lemma 3.3.13 The rst statement is immediate. Namely, the second Jacobi{Trudi identity for reads s = det(e 0 j +ij ) k i;j=1 = det(e (nk+i)uj ) k i;j=1 since 0 j +ij =nku j +j +ij = (nk +i)u j To prove the second statement, we proceed by induction on k. For k = 1 there is nothing to prove. For k> 1, consider the expansion of D = det(e nk+iuj (x 1 ;:::;x nk+i1 )) along the last row, i.e. D = k X j=1 e nuj (x 1 ;:::;x n1 )M j where M j is the signed minor of the matrix obtained by removing the last row and the jth column. By induction, M j depends exclusively on the variables x 1 ;:::;x nk , and M j = (1) k+j det(e nk+iu l (x 1 ;:::;x nk )) i=1;:::;k1 l=1;:::; b j;:::;k Applying the usual recursive relation for elementary symmetric functions e m (x 1 ;:::;x n ) = e m (x 1 ;:::;x n1 ) +x n e m1 (x 1 ;:::;x n1 ) we get D = k X j=1 e nuj (x 1 ;:::;x n1 )M j = = k X j=1 e nuj (x 1 ;:::;x n2 )M j +x n1 k X j=1 e n1uj (x 1 ;:::;x n2 )M j CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 99 Now we observe that k X j=1 e n1uj (x 1 ;:::;x n2 )M j = 0 since it describes the determinant of a matrix with two equal rows. By iterating this process we get D = k X j=1 e nuj (x 1 ;:::;x nk )M j = det(e nk+iuj (x 1 ;:::;x nk )) k i;j=1 Proof of Proposition 3.3.10 LetS2 Mat(nn;P n ) be the unipotent lower triangular matrix [S] ij = S sjsij = S c[j;i] = e ij (x 1 ;:::;x i1 ) for anyi>j. The elements# = (# 1 ;:::;# n ) satisfy# =S T !, where! = (! 1 ;:::;! n ). In particular, their wedge product can be written in terms of minors ofS. More specically, for every 2 (Z n 2 ) k , e # :=# 1 1 # n n = X D ! where D is the minor ofS corresponding to the rows identied by and the columns identied by . Proposition 3.3.15. For every ;2 (Z n 2 ) k , , S 1 = D . In particular, # = e # . Proof. Since is a Grassmannian permutation with descent atnk, it follows from Lemma 3.3.13 S = s (x 1 ;:::;x nk ) = det(e (nk+i)uj (x 1 ;:::;x nk )) k i;j=1 and S = det(e (nk+i)uj (x 1 ;:::;x nk )) k i;j=1 = det(e (nk+i)uj (x 1 ;:::;x nk+i1 )) k i;j=1 = D (k) CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 100 Moreover, since the elements # j are S n {invariant, so is e # . Hence the coecients D satisfy D =@ D (k) and therefore D =@ D (k) =@ S = S 1 This concludes the proof of Proposition 3.3.10. Remark 3.3.16. It follows from the discussion above that the Schubert exterior basis of ext n is more concisely described in terms of elementary functions. It will be convenient to reindex these elements. Henceforth, we will adopt the following notation e ! j = j1 X k=0 e k (x 1 ;:::;x nj+k )! n+1j+k =# nj+1 Dual Schubert polynomials Our second example of a basis for ext n relies on the notion of dual Schubert polynomials. Proposition 3.3.17 ([Man01] Proposition 2.5.7). There is a n -bilinear form on P n dened by (x;y) :=@ w0 (xy). With respect to this form the dual basis to the Schubert polynomials are given by S w = (1) `(ww0) w 0 (S ww0 ); w2 S n : (3.3.12) The dual Schubert polynomials are characterized by the following conditions: (i) S w0 = 1; (ii) for every u2 S n @ u S w = 8 > < > : S wu 1 if l(wu 1 ) =l(w) +l(u) 0 otherwise. CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 101 This follows directly from the characterization of the Schubert polynomials in 3.3.4 and from the relation w o @ u w 0 = (1) l(u) @ w0uw0 In particular, we get the following result, dualizing Proposition 3.3.9. Proposition 3.3.18. The elements p j = S w0c[j] , 1jn, are a solution of (3.3.10). In particu- lar, the elements # j =! j + X k>j S w0c[j;k] ! k dene an exterior basis of ext n . In 3.3.12, we showed that the Schubert polynomials involved in the exterior basis of ext n are elementary symmetric functions, namely, S c[j;k] = e kj (x 1 ;:::;x k1 ): The dual Schubert polynomials are, instead, naturally described by complete symmetric functions. By denition, we have S w0c[j;k] = (1) kj w 0 (S w0c[j;k]w0 ) = (1) kj w 0 (S c[nk+1;nj+1] 1) sincew 0 c[j;k]w 0 =s nj s nk+1 . The permutationc[nk+1;nj+1] 1 is still a Grassmannian permutation, whose unique descent is atnk + 1 and whose partition is conjugate to that ofc[j;k]. Therefore S c[nk+1;nj+1] 1 = h kj (x 1 ;:::;x nk+1 ) and S w0c[j;k] = (1) kj h kj (x k ;:::;x n ): In particular, the relation @ c[k] S w0c[j] = S w0c[j;k] reads @ c[k] ((1) nj h nj (x n )) = (1) kj h kj (x k ;:::;x n ) CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 102 providing a dierent proof of [AH15, Prop. 5.4]. As in the Schubert case, one observes that the dual Schubert polynomials give a solution of (3.3.4). Namely, one can set p = S w0 . Then @ S w0 = S w0 1 , @ S w0 = 1 and @ i S w0 1 = 0 for every and i62D . It follows that there are elements in ext n # =! + X S w0 1 ! which satisfy, in analogy with 3.3.10, # = (# 1 ) 1 (# n ) n . Remark 3.3.19. It follows from the discussion above that the dual Schubert exterior basis of ext n is concisely described in terms of complete functions. As before, it will be convenient to reindex these elements. Henceforth, we will adopt the following notation h ! j = j1 X k=0 (1) k h k (x n+1j+k ;:::;x n )! n+1j+k =# nj+1 A family of bases for ext n We now describe a collection of bases of ext n which interpolates between the Schubert basis 3.3.4 (described in terms of elementary symmetric functions) and the dual Schubert basis 3.3.4 (described in terms of complete symmetric functions). Recall that the elementary symmetric functions satisfy the relation e j (x 1 ;:::;x n1 ) = j X l=0 (1) jl x jl n e l (x 1 ;:::;x n ) For every 0rnj, set p (r) j =(1) r x r n e njr (x 1 ;:::;x n1 ) =(1) r x r n njr X l=0 (1) njrl x njrl n e l (x 1 ;:::;x n ) CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 103 = njr X l=0 (1) njl h njl (x n )e l (x 1 ;:::;x n ) Proposition 3.3.20. For every choice ofr, the elementsp (r) j , 1jn, are a solution of (3.3.10). In particular, the elements # (r) j = X kj @ c[k] (p (r) j )! k dene an exterior basis of ext n . Proof. Since @ c[k] (h njl (x n )) = 0 for every kj and l>kj, we have @ c[k] (p (r) j ) = kj X l=0 (1) kjl e l (x 1 ;:::;x n )h kjl (x k ;:::;x n ) Therefore, @ c[j] (p (r) j ) = 1 and @ i @ c[k] (p (r) j ) = 0 for every k>j and i6=k 1. The result follows. This basis interpolates between 3.3.4 and 3.3.4. Specically, for r = 0, p (0) j = e nj (x 1 ;:::;x n1 ) = S c[j] and we obtain the Schubert exterior basis 3.3.4. Instead, for r =nj, p (nj) j = (1) nj h nj (x n ) = S w0c[j] and we obtain the dual Schubert exterior basis 3.3.4. Indeed, more precisely, we have, for any 0rnj, p (r) j = S c[j+r] S w0c[nr] (3.3.13) Example 3.3.21. Set n = 3, then we have In particular, the Schubert exterior basis of ext 3 is # 1 =! 1 +x 1 ! 2 +x 1 x 2 ! 3 # 2 =! 2 + (x 1 +x 2 )! 3 # 3 =! 3 CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 104 w S w S w0w id 1 1 s 1 x 1 x 2 x 3 s 2 x 1 +x 2 x 3 s 1 s 2 x 1 x 2 x 2 3 s 2 s 1 x 2 1 x 2 x 3 s 1 s 2 s 1 x 2 1 x 2 x 2 x 2 3 Table 3.1: Example for n = 3. Instead, the dual Schubert exterior basis is # 1 =! 1 (x 2 +x 3 )! 2 +x 2 3 ! 3 # 2 =! 2 x 3 ! 3 # 3 =! 3 Other possible choices are obtained replacing # 1 or # 1 with # (1) 1 =! 1 +x 1 ! 2 (x 1 +x 2 )x 3 ! 3 corresponding to the choice p (1) 1 in 3.3.4. Other bases We conclude this section with two more examples. (i) Power functions. One can consider power symmetric polynomials and set p j = (1) nj p nj (x 1 ;:::;x n1 ) On the other hand, p j = (1) nj p nj (x 1 ;:::;x n1 ) = (1) nj p nj (x 1 ;:::;x n ) + (1) nj h nj (x n ) Therefore it simply gives back the description in terms of complete symmetric functions. (ii) Symmetrizers The easiest example, although computationally most expensive, is obtained by CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 105 full symmetrization of the exterior variables ! j , i.e. for every 1jn, set ! s j = 1 n! X 2Sn (! j ) 3.3.5 Combinatorial identities The following results give the relationship betweenfe w i g andfh w i g (see Remarks 3.3.16 and 3.3.19), where h ! j = j1 X k=0 (1) k h k (x n+1j+k ;:::;x n )! n+1j+k (3.3.14) e ! j = j1 X k=0 e k (x 1 ;:::;x nj+k )! n+1j+k (3.3.15) We use the following identity between elementary symmetric polynomials and complete homo- geneous symmetric polynomials to prove the next proposition: Lemma 3.3.22. e k (x 1 ; ;x nj+k ) = k X t=0 (1) k+t h kt (x nj+k+1 ; ;x n )e t (x 1 ; ;x n ): (3.3.16) Proof. Using standard facts about elementary and complete symmetric functions we have k X t=0 (1) t h kt (x nj+k+1 ; ;x n )e t (x 1 ; ;x n ) = k X t=0 (1) t h kt (x nj+k+1 ; ;x n ) t X a=0 e a (x 1 ; ;x nj+k )e ta (x nj+k+1 ; ;x n ) ! = k X a=0 k X t=a (1) t h kt (x nj+k+1 ; ;x n )e ta (x nj+k+1 ; ;x n )e a (x 1 ; ;x nj+k ) = k X a=0 ka X t 0 =0 (1) kt 0 h t 0(x nj+k+1 ; ;x n )e kat 0(x nj+k+1 ; ;x n )e a (x 1 ; ;x nj+k ) = (1) k k X a=0 e a (x 1 ; ;x nj+k ) ka X t 0 =0 (1) t 0 h t 0(x nj+k+1 ; ;x n )e kat 0(x nj+k+1 ; ;x n ) ! = (1) k k X a=0 e a (x 1 ; ;x nj+k ) ( a;k ) CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 106 = (1) k e k (x 1 ; ;x nj+k ): Proposition 3.3.23. For any 1jn, we have e w j = j1 X k=0 e k h w jk : Proof. Using the denition of h w j we have j1 X k=0 e k h w jk = e 0 (x 1 ; ;x n )h w j +e 1 (x 1 ; ;x n )h w j1 + +e j1 (x 1 ; ;x n )h w 1 =! n+1j + j1 X k=1 k1 X l=0 (1) kl h kl (x nj+k+1 ; ;x n )e l (x 1 ; ;x n ) ! ! nj+k+1 =! n+1j + X n+1j<kn e k(n+1j) (x 1 ; ;x k1 )! k = e w j where the third equality follows from Lemma 3.3.22 and the last step comes from a change of variables. 3.4 Solomon's Theorem 3.4.1 Superpolynomials and superinvariants Fix an integer n 1. Let x denote a set of formal even variables x 1 ;:::;x n , and let dx denote a set of formal odd variables dx 1 ;:::;dx n . Here \odd" means that these variables are assumed to anti-commute amongst themselves and square to zero. Thus, Q[x; dx] is short-hand for the superpolynomial ring Q[x; dx] :=Q[x 1 ;:::;x n ] Q ^ [dx 1 ;:::;dx n ]: We make this ring bigraded by declaring that deg(x i ) = (1; 0) and deg(dx i ) = (0; 1). CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 107 The symmetric group S n acts on Q[x; dx] by algebra automorphisms, dened by permuting indices: w(x i ) =x w(i) and w(dx i ) =dx w(i) . Note that this action preserves the bidegree. Theorem 3.4.1 (Solomon [Sol63]). For any family f =ff 1 ;:::;f n g of algebraically independent generators ofQ[x] Sn ,Q[x; dx] Sn =Q[f; df]: In particular, Q[x 1 ;:::;x n ;dx 1 ;:::;dx n ] Sn =Q[e 1 ;:::;e n ;de 1 ;:::;de n ]; where e i =e i (x 1 ;:::;x n ) is the i-th elementary symmetric polynomial, and de i 2Q[x; dx] is to be interpreted in the usual manner for functions: df := n X i=1 @f @x i dx i 8f2Q[x]: Note that deg(e i ) = (i; 0) and deg(de i ) = (i 1; 1). Remark 3.4.2. The mapping f7!df extends to a degree (1; 1) dierentialQ[x; dx]!Q[x; dx]. This is the usual exterior derivative on polynomial dierential forms. 3.4.2 Action of the extended nilHecke algebra Taking a cue from higher representation theory, we would like to consider divided dierence operators @ i acting on superpolynomials. Unlike in the case of ordinary polynomials, here it is necessary to introduce rational functions in the variablesx 1 ;:::;x n . So, let i :=x i x i+1 fori = 1;:::;n1, let 1 =f 1 1 ;:::; 1 n1 g, and consider the algebraQ[x; dx; 1 ]. Note that this algebra is bigraded, with deg((x i x i+1 ) 1 ) = (1; 0). We have the divided dierence operators@ i :Q[x; dx; 1 ]!Q[x; dx; 1 ] dened in the usual way @ i = 1s i x i x i+1 : It follows from Solomon's theorem that for any tuple f =ff 1 ;:::;f n g of algebraically independent generators ofQ[x] Sn the subalgebraQ[x; df]Q[x; dx; 1 ] is closed under the action of the divided dierence operators. CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 108 Consequently,Q[x; df] is a module over the extended nilHecke algebra. We wish to compare this module with the polynomial representation of the extended nilHecke algebra considered earlier. This representation can be described as follows. Let ! =f! 1 ;:::;! n g be a set of formal odd variables, with bidegree deg(! i ) = (ni; 1): The superpolynomial ringQ[x;!] admits an S n action via w(x i ) = x w(i) for all w2 S n , together with s j (! i ) = 8 > > < > > : ! i + (x i x i+1 )! i+1 if j =i, ! i otherwise. Note that the S n action preserves the bidegree. The actions ofQ[x] andQ[S n ] determines uniquely that of NH ext n . Note that the graded dimensions ofQ[x;!] andQ[x; df] coincide. Thus, it is natural to hope for a bidegree preserving isomorphism ofNH ext n -modulesQ[x;!] =Q[x; df]. Note that equivariance with respect to the NH ext n action is equivalent to linearity with respect toQ[x], together with equivariance with respect to S n . 3.4.3 Preliminary computations We say that a tuple p =fp 1 ;:::;p n gQ[x] is admissible if p j 2Q[x] Sn1S1 , deg(p j ) = nj, and @ c[j] p j 2Q for any j = 1;:::;n, where c[j] =s j s j+1 s n1 and c[n] = id. This implies, in particular, that the matrix P = [@ c[j] p i ] 1i;jn 2 Mat(n;Q[x]) is upper triangular and invertible. We introduce the following operators. For any ring R and any k = 1;:::;n 1, let k be the linear operator on Mat(mn;R) dened by k (A) ij = j;k+1 A ik CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 109 and let k be the linear operator on Mat(nm;R) dened by a k (A) ij = i;k A k+1;j : The following lemma gives a characterization of admissible tuples in terms of the corresponding matrices, obtained through divided dierence operators. Lemma 3.4.3. (i) If p =fp 1 ;:::;p n g is an admissible tuple, then P satises@ k (P) = k (P) for anyk = 1;:::;n 1, where the action of the divided dierence operator is dened entrywise. (ii) For any invertible Q = [q ij ]2 Mat(n;Q[x]) such that @ k (Q) = k (Q) for k = 1;:::;n 1, and deg(q ij ) =ji, the tuple q =fq 1n ;:::;q nn g is admissible and Q ij =@ c[j] q in . Proof. (i) follows immediately from the fact that p i 2Q[x] Sn1S1 and therefore @ k @ c[j] p i = j;k+1 @ c[k] p i : Let now Q be a solution of @ k (Q) = k (Q). Then, for any k = 1;:::;n 2, @ k q in = 0 and q in 2Q[x] Sn1S1 , i = 1;:::;n. Moreover, q i;k =@ k q i;k+1 = =@ k @ k+1 @ n1 q in =@ c[k] q in : Finally, since deg(q ij ) = ji and Q is invertible, it follows that @ c[j] q jn 2 Q . Therefore q = fq 1n ;:::;q nn g is admissible. This proves (ii). We now consider the following situation. Let =f 1 ;:::; n g; =f 1 ;:::; n g be two sets of algebraically independent elements inQ[x; dx] such that deg( i ) = (ni; 1) = deg( i ),i = 1;:::;n, and let P2 Mat(n;Q[x]) be the invertible matrix dened by the relation = P: (3.4.1) a In other words, k gives back the kth column of A in (k + 1)th position, while k gives back the (k + 1)th row of A in position k. CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 110 Note that, necessarily, deg(p ij ) =ji. Lemma 3.4.4. Any two of these equations imply the third: (a) @ k (P) = k (P); (b) @ k () = 0; (c) @ k () = k (). Proof. We rst show that if (a) holds, then (b) and (c) are equivalent, that is, if@ k (P) = k (P), then @ k () = 0 () @ k = k (): (3.4.2) Namely, sinces k ( k (P)) =s k (@ k (P)) =@ k (P) = k (P), one checks easily that k (P) =s k (P) k (), where the action ofs k is dened entrywise as in the case of@ k . Now, the application of@ k to (3.4.1) gives @ k () =@ k (P) +s k (P)@ k () = = k (P) +s k (P)@ k () = =s k (P) ( k () +@ k ()) Therefore, (3.4.2) follows from the invertibility of P. In particular, we proved that (a) and (b) imply (c), and (a) and (c) imply (b). It remains to show that (b) and (c) imply (a), that is, if @ k () = 0 and @ k () = k (), then @ k (P) = k (P). In this case, the application of @ k to (3.4.1) gives 0 =@ k (P) +s k (P)@ k () =@ k (P)s k (P) k () Denote by P 1 ;:::;P n the column vectors of P. Since the component of =f 1 ;:::; n g are alge- braically independent overQ[x], the equation @ k (P) =s k (P) k () implies @ k P i = i;k+1 s k (P k ) CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 111 and therefore @ k P = k (P). 3.4.4 NH ext n {equivariant isomorphisms Let f =ff 1 ;:::;f n g be a set of algebraically independent generators ofQ[x] Sn , with deg(f i ) =ni, p =fp 1 ;:::;p n gQ[x] an admissible tuple and set P = [@ c[j] p i ] i;j=1;:::;n . Proposition 3.4.5. For any choice of f and p, there is a uniqueQ[x]{linear algebra homomorphism J f p :Q[x;!]!Q[x; dx; 1 ] dened by the relation df = P J f p (!). Moreover, J f p is injective, NH ext n {equivariant, and degree preserving. Proof. Since p is admissible, the matrix P is invertible and the algebra homomorphism J f p is uniquely determined by the condition df = PJ f p (!) and linearity inQ[x]. The injectivity of J f p follows from the invertibility of P and the algebraic independence of the elements f =ff 1 ;:::;f n g and df =fdf 1 ;:::;df n g. The S n {equivariance follows from Q[x]{linearity and Lemmas 3.4.3, 3.4.4. Namely, since p is admissible, it follows from Lemma 3.4.3 that @ k (P) = k (P). Then, since df = PJ f p (!) and @ k (df) = 0, it follows from Lemma 3.4.4 that @ k (J f p (!)) = k (J f p (!)), which is equivalent to s i (J f p (! j )) = J f p (! j ) + ij (x i x i+1 )J f p (! i+1 ) and implies the S n {equivariance of J f p . The NH ext n {equivariance follows. Finally, the fact that J f p preserves the degree is a straightforward check. The construction of the homomorphism J f p allows us to compare the description of the S n { invariants inQ[x;!] from Theorem 3.3.3 and that of the S n {invariants inQ[x; dx] from Solomon's Theorem. We obtain the following Corollary 3.4.6. The homomorphisms J f p restricts to a canonical identication of S n {invariants. CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 112 More specically, there is a commutative diagram Q[x;!] J f p Q[x;! p ] P oo Q[x;! p ] Sn ? _ oo Q[f;! p ] Q[x; dx; 1 ] Q[x; df] ? _ oo Q[x; df] Sn ? _ oo Q[f; df] where P denotes the change ofQ[x]{basis dened by! p = P! and the vertical arrows send! p to df. 3.4.5 Example Let h =fp 1 ;:::;p n g be the admissible tuple with p j = (1) nj h nj (x n ), and let H be the corre- sponding matrix. In particular, H ij =@ c[j] p i = (1) ji h ji (x j ;:::;x n ) It is easy to see that the homomorphism J f h is dened by J f h (!) = Qdf where Q ij = e ji (x i+1 ;:::;x n ) Similarly, let e =fp 1 ;:::;p n g be the admissible tuple with p j = e nj (x 1 ;:::;x n1 ), and let E be the corresponding matrix. In particular, E ij =@ c[j] p i = e ji (x 1 ;:::;x j1 ) CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 113 and the homomorphism J f e is dened by a J f e (!) = e Qde where e Q ij = (1) ji h ji (x 1 ;:::;x i ) 3.5 Dierentials In this section we show that the dierentiald N on NH ext n dened in section 3.2.3 restricts to the ring of extended symmetric functions ext n . We identify the resulting DG-algebra as the Koszul complex associated to a certain regular sequence of symmetric polynomials in n , whose cohomology is isomorphic to the cohomology ring of a Grassmannian. We also dene new deformed dierentials d N on NH ext n in section 3.5.3. The deformed dierentials also restrict to ext n and the resulting cohomology of ( ext n ;d N ) is related to GL(N)-equivariant cohomology of a Grassmannian. The reader may wish to recall the grading conventions from section 3.2.3. 3.5.1 The standard dierential Recall that NH ext n admits a dierential d N for each Nn 1, dened by d N (! i ) = (1) i h Ni+1 (x 1 ;:::;x i ) d N (x i ) = 0 d N (@ i ) = 0 for all i, together with the Leibniz rule. Consequently, d N is linear with respect to the subalgebra NH n NH ext n . Remark 3.5.1. With respect to the bigradings (3.2.1), the dierential d N is homogeneous with degree (2 2N;1). a Both computations follow easily from the relation between the generating series of elementary and complete functions. More specically, for j >i, one has 0 @ X k0 (1) k t k h k (x j ;:::;xn) 1 A 0 @ X k0 t k e k (x i+1 ;:::;x j ;:::;xn) 1 A = j1 Y l=i+1 (1 +tx l ) In particular, comparing the coecients of t ji , we get j X k=i (1) jk h jk (x j ;:::;xn)e ki (x i+1 ;:::;xn) = 0 which implies that the entries of H 1 are the polynomials e ji (x i+1 ;:::;xn). Similarly for E. CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 114 The following states that ext n is a DG-subalgebra of NH ext n in a natural way. Proposition 3.5.2. The dierential d N restricts to a dierential on ext n NH ext n . Proof. The subset ext n = Z(NH ext n ) NH ext n can be characterized as the set consisting of those elementsz2 NH ext n such that [@ i ;z] = 0 for all divided dierence operators@ i 2 NH ext n . On the other hand d N is NH n -linear, so [@ i ;d N (z)] =d N ([@ i ;z]) = 0 if [@ i ;z] = 0. Another way to prove this statement is as follows: Proof. By Theorem 3.3.3 we know that ext n ' n V [(]! s 1 ;:::;! s n ) and v2 ext n if and only if v = P f (i1;;i k ) ! s i1 ! s i k then d N (v) = X d N (f (i1;;i k ) ! s i1 ! s i k ) Since d N is linear, it is enough to consider dierential of f! s i1 ! s i k for any f2 n and for any ktuple (i 1 ; ;i k ) where kn and 1i 1 <<i k n. d N (f! s i1 ! s i k ) =d N (f)! s i1 ! s i k + (1) deg(f) fd N (! s i1 ! s i k ) =fd N (! s i1 ! s i k ) =f d N (! s i1 )! s i2 ! s i k + (1) deg(! s i 1 ) ! s i1 d N (! s i2 ! s i k ) : Therefore it is enough to check d N (# i ) where # i = ni X j=0 (1) j h j (x i+j ; ;x n )! i+j CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 115 as dened in Section 3.3.4. Then d N (# i ) = ni X j=0 (1) j d N (h j (x i+j ; ;x n ))! i+j + (1) deg(hj (xi+j;;xn)) h j (x i+j ; ;x n )d N (! i+j ) = ni X j=0 (1) j h j (x i+j ; ;x n )d N (! i+j ) = ni X j=0 (1) j h j (x i+j ; ;x n )(1) i+j h Nij+1 (x 1 ; ;x i+j ) =(1) i h Nn+1 (x 1 ; ;x n ) where the last equality follows from lemma 3.5.4. Therefore, d N (# i )2 ext n and ( ext n ;d N ) is a DG-subalgebra. Example 3.5.3. Let us consider the dierential d N of h ! j . We will see that d N (h ! j ) lands in ext n , by direct computation. Recall from Remark (3.3.19) that for n = 3 h ! 1 =! 3 ; h ! 2 =! 2 x 3 ! 3 ; h ! 3 =! 1 (x 2 +x 3 )! 2 +x 3 2 ! 3 : Then the dierentials are computed as follows. d N (h ! 1 ) =d N (! 3 ) = (1) 3 h N2 (x 1 ;x 2 ;x 3 ) d N (h ! 2 ) =d N (! 2 x 3 ! 3 ) = h N1 (x 1 ;x 2 ) +x 3 h N2 (x 1 ;x 2 ;x 3 ) = h N1 (x 1 ;x 2 ;x 3 ): The last equality comes from the following observation: f(a;b;c)ja +b +c =N 1g =f(a;b; 0)ja +b =N 1g[f(a;b;c)ja +b +c =N 1;c 1g CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 116 Similarly d N (h ! 3 ) =d N (! 1 (x 2 +x 3 )! 2 +x 2 3 ! 3 ) =x N 1 (x 2 +x 3 )h N1 (x 1 ;x 2 )x 2 3 h N2 (x 1 ;x 2 ;x 3 ) =h N (x 1 ;x 2 ;x 3 ): Similar to the above argument, the last equality follows from the observation: f(a;b;c)ja +b +c =Ng =f(a; 0; 0)ja =Ng[f(a;b; 0)ja +b =N; b 1g [f(a;b; 1)ja +b =N 1g[f(a;b;c)ja +b +c =N; and c 2g: Before we compute d N (h ! j ) in general, we need the following result on symmetric functions. Lemma 3.5.4. Let h i (x j ; ;x n ) denote the complete homogeneous symmetric polynomial of degree i in variables x j ; ;x n , for 1jn. Then for any 1in and N2N h Ni+1 (x 1 ; ;x n ) = ni X j=0 h Nij+1 (x 1 ; ;x i+j )h j (x i+j ; ;x n ): (3.5.1) Proof. For any 0jni, 1in, and N2N h Nij+1 (x 1 ; ;x i+j )h j (x i+j ; ;x n ) = 0 @ X b1++bi+j =Nij+1 x b1 1 x bi+j i+j 1 A 0 @ X ai+j ++an=j x ai+j i+j x an n 1 A = X b1++bi+j =Nij+1 X ai+j ++an=j x b1 1 x bi+j +ai+j i+j x ai+j+1 i+j+1 x an n = j X k=0 0 @ X b1++bi+j =Nij+1 0 @ X ai+j ++an=jk x b1 1 x bi+j +k i+j x ai+j+1 i+j+1 x an n 1 A 1 A The exponent of each monomial in above sum is anntuple (b 1 ; ;b i+j +k;a i+j+1 ; ;a n ) where b 1 + +b i+j =Nij + 1; CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 117 a i+j+1 + +a n =jk; and a i+j =k for any 0kj: Asj varies in the range 0jni these exponents exhaust uniquely all monomials appearing in h Ni+1 (x 1 ; ;x n ). 3.5.2 Koszul complex Let R be a commutative ring, and let a 1 ;:::;a r 2 R be given elements. The Koszul complex associated to (a 1 ;:::;a r ) is the DG algebra R ^ [ 1 ;:::; r ] withR-linear dierential uniquely characterized byd( i ) =a i together with the graded Leibniz rule. For the purposes of the Leibniz rule, the grading places R in homological degree zero, and each i in homological degree1. Proposition 3.5.5. As a DG-algebra, ext n is isomorphic to the Koszul complex associated to (1) i h Ni+1 2 n (1in). Proof. By Theorem 3.3.3 we know that ext n ' n V [! s 1 ;:::;! s n ], where! s j =! s j (p j ) are determined by any choice of p j 2Q[x 1 ;:::;x n ] Sn1S1 such that @ j @ j+1 @ n1 (p j ) = 1. For the purposes of computing the dierential, it is especially convenient to work with the choice of p j as constructed in 3.3.4. In this case the resulting elements ! s i are given by # i := ni X j=0 (1) j h j (x i+j ; ;x n )! i+j : We know that the dierential d N is linear with respect to the subalgebra n (this follows from d N (x i ) = 0 and the Leibniz rule), hence to prove the Proposition we need only show that d N (# i ) = CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 118 (1) i h Ni+1 (x 1 ; ;x n ). Compute: d N (# i ) = ni X j=0 (1) j h j (x i+j ; ;x n )d N (! i+j ) = ni X j=0 (1) j h j (x i+j ; ;x n )(1) i+j h Nij+1 (x 1 ; ;x i+j ) =(1) i h Ni+1 (x 1 ; ;x n ) where the last equality follows from Lemma 3.5.4. A sequence of elements a = (a 1 ;:::;a r )2R is called a regular sequence if a 1 is not a zero divisor. a i is not a zero divisor in R=ha 1 ;:::;a i1 i for all 2in. If a is regular, then the associated Koszul complexK(a) has cohomology only in degree zero, where it is isomorphic to R=ha 1 ;:::;a r i. Said dierently, if a is a regular sequence then the canonical projection K(a)!R=ha 1 ;:::;a r i is a quasi-isomorphism. Corollary 3.5.6. The DG-algebra ( ext n ;d N ) is quasi-isomorphic to the cohomology ringH (Gr(n;N)). Proof. The sequence h N ;h N1 ;:::;h Nn+1 2 n is a regular sequence, see for example [Wu14, Proposition 7.2]. Thus, the cohomology of the associated Koszul complex is isomorphic to the quo- tient n =hh N ;h N1 ;:::;h Nn+1 i, which is known to be isomorphic to n =hh Nn+1 i =H (Gr(n;N)). 3.5.3 Deformed dierentials Deformed cyclotomic quotients The cyclotomic quotients of the nilHecke algebra, and KLR algebras more generally, admit deforma- tions called deformed cyclotomic quotients dened in [Web10]. For us the most relevant reference is [RW15, Section 3.2]. CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 119 Let 1 ;:::; N 2C be given, and let denote the root multiset consisting of pairwise distinct complex numbers 1 ;:::; ` corresponding to the roots of the polynomial P (x) =x N + N X j=1 j x Nj ; (3.5.2) with multiplicities N 1 ;:::;N ` . For each N > 0 dene the deformed cyclotomic ideal I N associated to as the ideal of NH n dened by I N := * N X j=0 j x Nj 1 + ; i 2C (3.5.3) where we take 0 = 1. We dene the deformed cyclotomic quotient NH n := NH n =I N : In [RW15, Section 3.2] it is shown that the deformed cyclotomic quotient rings NH n are iso- morphic to matrix rings of size n! with coecients in the GL(N)-equivariant cohomology ring H GL(N) (Gr(n;N)) with equivariant parameters equal to = ( 1 ; 2 ;:::; N ). We denote this spe- cialization by H n . If the parameters are left generic, then the center of the deformed cyclotomic quotient is just the GL(N)-equivariant cohomology itself [Wu12, Theorem 2.10]. Theorem 3.5.7 (Theorem 13 [RW15]). There is an algebra isomorphism H n = M P n j =n 0njN ` O j=1 H (Gr(n j ;N j )): We will realize both the deformed cyclotomic quotient NH n and the ringsH n within the context of the extended nilHecke algebra. For these realization we make use of the following lemma. Lemma 3.5.8. The following identities hold in NH n . 1. For any 1in, P N j=0 j x Nj i = 0. CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 120 2. For any mN, Nm+1 X j=0 j X P ai=(Nm+1j) x a1 1 x a2 2 :::x am m = Nm+1 X j=0 j h Nm+1j (x 1 ; ;x m ) = 0: Proof. The rst claim is proven by induction. Namely, for any 0in1, we prove the following. (A1) i For any y2N, P N j=0 j x y+(Nj) i+1 @ i = P N j=0 j @ i x y+(Nj) i+1 . (A2) i For any y2N, P N j=0 j x y+(Nj) i+1 @ i = 0. (A3) i P N j=0 j x Nj i+1 = 0. Recall that by denition @ 0 = 1. The case i = 0 holds by construction. Assume now (A1) i1 , (A2) i1 , (A3) i1 , with i> 0. One has N X j=0 j x y+Nj i+1 @ i = N X j=0 j @ i x y+Nj i N X j=0 j h y+Nj1 (x i ;x i+1 ) = = N X j=0 j h y+Nj1 (x i ;x i+1 ) = = N X j=0 j x y+Nj i @ i N X j=0 j h y+Nj1 (x i ;x i+1 ) = = N X j=0 j @ i x y+Nj i+1 where the rst and fourth equalities follow by (3.1.2), the second and third ones follow by (A3) i1 . This proves (A1) i . Then, (A2) i holds, since N X j=0 j x y+Nj i+1 @ i = N X j=0 j x y+Nj i+1 @ i x i+1 @ i = = N X j=0 j @ i x y+Nj+1 i+1 @ i = = N X j=0 j x y+Nj+1 i+1 @ 2 i = 0 where the rst equality follows from@ i =@ i x i+1 @ i , the second and third ones from (A1) i . Finally, CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 121 (A3) i holds, since N X j=0 j x y+Nj i+1 = N X j=0 j x y+Nj i+1 @ i x i N X j=0 j x y+Nj+1 i+1 @ i = 0 where the rst equality follows from the nilHecke relations (3.1.1) and the second one from (A2) i . This proves the rst claim. The second claim is similarly proven by induction. Using (3.1.1), we have N X j=0 j @ i x Nj i N X j=0 j x Nj i @ i = N1 X j=0 j 0 @ X a+b=y+(Nj)1 x a i x b i+1 1 A : The induction step is identical to Proposition 2.8 in [HL10]. Deformed dierentials Let denote the root multiset corresponding to the roots and multiplicities of the polynomial (3.5.2). To each dene a dierential d N on NH ext n , which we call deformed dierential, by d N (@ i ) = 0, d N (x i ) = 0, and d N (! i ) = P Ni+1 j=0 (1) i+1 j h Ni+1j (x 1 ; ;x i ). (3.5.4) Note that the deformed dierential d N is homogeneous of degree -1 with respect to the homo- logical grading deg h , but is in general not homogeneous with respect to deg q . Thus, we will regard (NH ext n ;d N ) as only a singly graded object (via deg h ). Proposition 3.5.9. The map d N satises the relations 1. @ i d N (! i+1 ) =d N (! i+1 )@ i 2. @ i d N (! i ) +d N (! i+1 )x i+1 @ i =d N (! i )@ i +@ i x i+1 d N (! i+1 ) for all 1in. Proof. The rst identity holds since d N (! i+1 ) is symmetric in x i andx i+1 . For the second identity, we show that d N (! i+1 )x i+1 @ i @ i x i+1 d N (! i+1 ) =d N (! i )@ i @ i d N (! i ) CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 122 One has d N (! i+1 )x i+1 @ i = Ni X j=0 (1) i+2 j (h Nij (x 1 ; ;x i+1 )x i+1 @ i ) = Ni X j=0 (1) i+2 j 0 @ X a+b=Nij h a (x 1 ; ;x i1 )h b (x i ;x i+1 )x i+1 @ i 1 A = Ni X j=0 (1) i+2 j 0 @ X a+b=Nij h a (x 1 ; ;x i1 ) X k+`=b x k i x `+1 i+1 @ i 1 A = Ni X j=0 (1) i+2 j 0 @ X a+b=Nij h a (x 1 ; ;x i1 )x b+1 i+1 @ i 1 A + Ni X j=0 (1) i+2 j 0 @ X a+b=Nij h a (x 1 ; ;x i1 ) X k 0 +` 0 =b1 x k 0 +1 i x ` 0 +1 i+1 @ i 1 A = Ni X j=0 (1) i+2 j 0 @ X a+b=Nij h a (x 1 ; ;x i1 )@ i x b+1 i 1 A Ni X j=0 (1) i+2 j 0 @ X a+b=Nij h a (x 1 ; ;x i1 )h b (x i ;x i+1 ) 1 A + Ni X j=0 (1) i+2 j 0 @ X a+b=Nij h a (x 1 ; ;x i1 ) X k 0 +` 0 =b1 x k 0 +1 i x ` 0 +1 i+1 @ i 1 A = Ni X j=0 (1) i+2 j 0 @ X a+b=Nij h a (x 1 ; ;x i1 )@ i x b+1 i 1 A Ni X j=0 (1) i+2 j h Nij (x 1 ;:::;x i+1 ) + Ni X j=0 (1) i+2 j 0 @ X a+b=Nij h a (x 1 ; ;x i1 ) X k 0 +` 0 =b1 x k 0 +1 i x ` 0 +1 i+1 1 A @ i where the fth equality follows by applying (3.1.2) to the rst summand. A similar computation gives @ i x i+1 d N (! i+1 ) = Ni X j=0 (1) i+2 j X a+b=Nij h a (x 1 ; ;x i1 )@ i x b+1 i+1 CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 123 + Ni X j=0 (1) i+2 j 0 @ X a+b=Nij h a (x 1 ; ;x i1 ) X k 0 +` 0 =b1 x k 0 +1 i x ` 0 +1 i+1 1 A @ i Therefore, d N (! i+1 )x i+1 @ i @ i x i+1 d N (! i+1 ) = Ni X j=0 (1) i+2 j X a+b=Nij h a (x 1 ; ;x i1 )@ i (x b+1 i x b+1 i+1 ) + Ni X j=0 (1) i+1 j h Nij (x 1 ; ;x i+1 ) On the other hand, one has d N (! i )@ i @ i d N (! i ) = Ni+1 X j=0 (1) i+1 j h Ni+1j (x 1 ; ;x i )@ i @ i Ni+1 X j=0 (1) i+1 j h Ni+1j (x 1 ; ;x i ) = Ni+1 X j=0 (1) i+1 j 0 @ X a+b=Ni+1j h a (x 1 ; ;x i1 )h b (x i ) 1 A @ i @ i Ni+1 X j=0 (1) i+1 j 0 @ X a+b=Ni+1j h a (x 1 ; ;x i1 )h b (x i ) 1 A = Ni+1 X j=0 (1) i+1 j 0 @ h Ni+1j (x 1 ; ;x i1 )@ i + X a+b 0 =Nij h a (x 1 ; ;x i1 )x b 0 +1 i @ i 1 A @ i Ni+1 X j=0 (1) i+1 j 0 @ h Ni+1j (x 1 ; ;x i1 ) + X a+b 0 =Nij h a (x 1 ; ;x i1 )x b 0 +1 i 1 A = Ni+1 X j=0 (1) i+1 j 0 @ X a+b 0 =Nij h a (x 1 ; ;x i1 )@ i x b 0 +1 i+1 1 A + Ni+1 X j=0 (1) i+1 j 0 @ X a+b 0 =Nij h a (x 1 ; ;x i1 )h b 0(x i ;x i+1 ) 1 A Ni+1 X j=0 (1) i+1 j 0 @ X a+b 0 =Nij h a (x 1 ; ;x i1 )@ i x b 0 +1 i 1 A = Ni+1 X j=0 (1) i+1 j h Nij (x 1 ; ;x i+1 ) CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 124 + Ni+1 X j=0 (1) i+1 j 0 @ X a+b 0 =Nij h a (x 1 ; ;x i1 )@ i (x i+1 b 0 +1 x b 0 +1 i ) 1 A where the fourth equality follows again from (3.1.2). Finally, since there is no contribution for j =Ni + 1, one has d N (! i )@ i @ i d N (! i ) = Ni X j=0 (1) i+2 j 0 @ X a+b 0 =Nij h a (x 1 ; ;x i1 )@ i (x i b 0 +1 x b 0 +1 i+1 ) 1 A + Ni X j=0 (1) i+1 j h Nij (x 1 ; ;x i+1 ) and the second identity follows. Corollary 3.5.10. The deformed dierential d N denes a degree1 dierential on NH ext n . Proof. The only nontrivial relations to verify are proven in Proposition 3.5.9. Theorem 3.5.11. The DGalgebra (NH ext n ;d N ) is quasi-isomorphic to deformed cyclotomic quo- tient of the nilHecke algebra NH n = NH n =h( P N j=0 j x Nj 1 )i. Proof. The statement follows immediately from the identity (2) of Lemma 3.5.8. That is, since Nm+1 X j=0 j h Nm+1j (x 1 ; ;x m ) = 0 in NH n , the same holds for the image of d N (! i ) in NH n . Proposition 3.5.12. For each N > 0, the pair ( ext n ;d N ) is a DG-subalgebra of (NH ext n ;d N ). Proof. This is immediate since the dierential d N acting on h ! i can be expressed as a linear combi- nation of undeformed dierentials, each of which preserves the ring ext n . Theorem 3.5.13. The DG-algebra ( ext n ;d N ) is quasi-isomorphic to the ringH n from Theorem 3.5.7. Proof. This follows from [RW15, Lemma 11] and Proposition 3.5.5. CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 125 3.5.4 Categorication Let f denote the positive part U + (sl 2 ) of the quantized universal enveloping algebra of sl 2 . This Q(q)-algebra is a polynomial ring in the generator E. This algebra is N-graded with E in degree 2. We equip the tensor product f f with the twisted algebra structure (E a E b )(E c E d ) =q 2cd E a E c E b E d : The algebra f contains a subring A f which is the Z[q;q 1 ]-lattice generated by all products of quantum divided powers E (n) := E n [n]! : (3.5.5) Hence, a categorication of A f amounts to identifying objectsE (n) andE n in a graded category and lifting the divided power relation (3.5.5) to an explicit isomorphism E n = M [n]! E (n) =E (n) hn 1iE (n) hn 3iE (n) h1ni: (3.5.6) The extended nilHecke algebra has been studied in connection with Verma modules by Naisse and Vaz [NV16, NV17b]. Here we show that the results from the previous section allow us to dene a categorication of A f, and in particular, dene categorications of quantum divided powers. For this it suces only to consider only the quantum grading on the extended nilHecke algebra, as we will do throughout this section, regarding NH ext n as a Z-graded algebra. Consider the Z- graded ring NH ext := M n0 NH ext n ; and denote by NH ext gmod the category of projective graded NH ext -modules. Recall from Proposi- tion 3.3.7 the isomorphism NH ext n = Mat((n) ! q 2 ; ext n ). One can easily show that e n = x @ w0 is the minimal idempotent projecting onto the lowest degree column of NH ext n . The graded module NH ext n e n is the unique indecomposable projective of NH ext n up to isomorphism and grading shift. The regular representation then decomposes inton! isomorphic copies of NH ext n e n . Taking gradings into account, CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 126 if we dene E (n) := NH ext n e n hn(n 1)=2i; E n := NH ext n ; then we have an isomorphism of graded projective left modules E n := NH ext n = M [n]! NH ext n e n =: M [n]! E (n) : Hence, we have proven the following. Proposition 3.5.14. There is an isomorphism ofA-modules : A f!K 0 (NH ext ) (3.5.7) sending E (n) to the class of the indecomposable projective moduleE (n) . There are inclusions of graded rings n;m : NH ext n NH ext m ! NH ext n+m (3.5.8) given diagrammatically by placing diagrams side-by-side with those in NH ext n appearing above NH ext m . In order to make this inclusion graded, it is necessary to adjust the gradings of the odd generators in NH ext m by an appropriate amount, as in Remark 3.2.1. In the notation of the aforementioned remark, the above map should be written n;m : (NH ext n ) (0) (NH ext m ) (n) ! (NH ext n+m ) (0) : These inclusions give rise to induction and restriction functors Ind n;m : (NH ext n ) (0) (NH ext m ) (n) gmod! (NH ext n+m ) (0) gmod; Res n;m : (NH ext n+m ) (0) gmod! (NH ext n ) (0) (NH ext m ) (n) gmod; By the basis theorem 3.3.3 for NH ext n+m it follows that the super module NH ext n+m is a free graded left super (NH ext n ) (0) (NH ext m ) (n) -module. A basis is given by the crossing diagrams in NH ext n+m CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 127 corresponding to the minimal representative of a left S n S m -coset inS n+m , see for example [KL09, Proposition 2.16]. It follows that Res n;m takes projectives to projectives, and therefore descends to a map in the Grothendieck group. Similarly, by a version of the Mackey induction-restriction theorem it follows that Ind n;m also sends projectives to projectives. At the level of Grothendieck groups we have [Ind n;m ]: K 0 (NH ext n ) (0) (NH ext m ) (n) !K 0 ((NH ext n+m ) (0) ); [Res n;m ]: K 0 ((NH ext n+m ) (0) )!K 0 (NH ext n ) (0) (NH ext m ) (n) : Since K 0 ((NH ext n ) (m) ) is canonically isomorphic to K 0 ((NH ext n ) (0) ) = K 0 (NH ext n ), these maps induce maps [Ind n;m ]: K 0 (NH ext n ) K 0 (NH ext m )!K 0 (NH ext n+m ); [Res n;m ]: K 0 (NH ext n+m !K 0 (NH ext n ) K 0 (NH ext m ): Summing over all n;m2 Z 0 these functors induce maps [Ind]: K 0 NH ext K 0 NH ext !K 0 NH ext ; [Res]: K 0 NH ext !K 0 NH ext K 0 NH ext : Just as in the case of the nilHecke algebra, see [KL09], induction and restriction equip A f with the structure of a twisted bialgebra and we have the following result. Theorem 3.5.15. The isomorphism : A f!K 0 (NH ext ) (3.5.9) is an isomorphism of twisted bialgebras. Remark 3.5.16. In [NV17b, Section 3.6] they independently considered a related construction where they sum the algebras (NH ext n ) (t) over both n;t2 Z. They then take the sum over t2 Z of CHAPTER 3. A DG-EXTENSION OF SYMMETRIC FUNCTIONS 128 induction and restriction functors Ind (t) n;m : (NH ext n ) (t) (NH ext m ) (n+t) gmod! (NH ext n+m ) (t) gmod; Res (t) n;m : (NH ext n+m ) (t) gmod! (NH ext n ) (t) (NH ext m ) (n+t) gmod: At the level of Grothendieck rings, this corresponds to a direct sum over t2 Z many copies of A f. They regard this as a copy of the positive part of sl(2) inside the Beilinson-Lusztig-MacPherson idempotent form of the quantum group, since their construction eectively includes idempotents indexed by the weight lattice t2 Z. Chapter 4 Conclusions and Future Work In the rst part of this thesis, we studied dierentials on the odd 2-category U dened by Brundan and Ellis [BE17b]. We rst classied derivations on this category and then proved that the odd 2-category U admits dierentials which give rise to a fantastic ltration. Then we conjecture that the Grothendieck ring of this DG odd 2-category (U;@) is the small quantum groups at 4th root of unity, which will be given in [EL18]. Dierentials on the odd 2-category is still an ongoing project that we are working on. We are investigating in detail for the possible relations between Khovanov homology and knot Floer homology. While working on this project, we realized that taking Grothendieck ring of a DG-category is a very hard problem. Which can be possible if the category, one is working with satises certain properties so that some known tools can be used. In our case it was the existence of fantastic ltration. Therefore, we only considered the dierentials which give rise to a fantastic ltration. If there are more tools for taking the Grothendieck ring, one can consider dierentials on the odd 2-category U which can be dened using the generators and the dening relations of the category, and as a result the DG-category could be categorication of more interesting algebras. One of the possible projects is to dene a thick calculus for the odd 2-category U as dened for Lauda's 2-category in [KLMS12]. Another one is to decategorify this odd 2-category U using trace decategorication as described in Section 1.1. It was shown that trace decategorication of 129 CHAPTER 4. CONCLUSIONS AND FUTURE WORK 130 categoried quantum groups gives inegral idempotented version of current algebra, see [BHLZ14, BGHL]. This result further implies that representation theory of current algebra can be understood through studying the 2-representations of categoried quantum groups. Representation theory of current algebras is motivated by its relationship to representation theory of ane and quantum ane associated to a simple lie algebra g, see [CL07, CG05, BC15, CV14]. Moreover, how trace decategorication functor interacts with dierential in p-dg theory [EQ16b, EQ16a], with dierential on the odd nilHecke algebra [EQ16c] is another interesting problem. Simi- larly, Brundan's odd 2-category U which we equipped with a dierential, relation between trace and the dierential is not known. Another future work is to study the same problems and dierentials for sl n and other types. We only considered g = sl 2 , in this thesis and it can be extended to sl n and other lie algebras g, which is a harder problem. In the second part of the thesis, we studied the extended symmetric functions which invariant under the action of extended nilHecke algebra and classied them. Moreover, we dened a family of dierentials on them. 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Abstract (if available)

Abstract This thesis studies DG structures on categorified quantum groups. In the first part of the thesis, we equip Ellis and Brundan's version of the odd categorified quantum group for sl₂ with a differential and identify the Grothendieck ring of the associated dg-2-category with a categorification of sl₂ at a fourth root of unity. In the second part of the thesis we investigate analogs of symmetric functions arising from an extension of the nilHecke algebra defined by Naisse and Vaz. These extended symmetric functions form a subalgebra of the polynomial ring tensored with an exterior algebra. We define families of bases for this algebra and show that it admits a family of differentials making it a sub-DG-algebra of the extended nilHecke algebra. The ring of extended symmetric functions equipped with this differential is quasi-isomorphic to the cohomology of a Grassmannian. We also introduce new deformed differentials on the extended nilHecke algebra that when restricted makes extended symmetric functions quasi-isomorphic to GL(N)-equivariant cohomology of Grassmannians.

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Creator Egilmez, Ilknur (author)

Core Title DG structures on categorified quantum groups

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 07/16/2018

Defense Date 03/22/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag categorification,OAI-PMH Harvest,quantum algebra,quantum groups

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Lauda, Aaron (committee chair), Fulman, Jason (committee member), Ortega, Antonio (committee member)

Creator Email egilmez@usc.edu,ilknur.egilmez@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-15914

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