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A categorification of the internal braid group action of the simply laced quantum group

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A categorification of the internal braid group action of the simply laced quantum group

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Content A Categorification of the Internal Braid Group Action of the Simply Laced Quantum Group Laffite Lamberto-Egan Doctor of Philosophy (Mathematics) Faculty of the USC Graduate School University of Southern California 12-21-2016 Abstract ForagivenquantumgroupU q (g),Lusztigdefinesasetofautomorphismsτ ′ i,e ,τ ′′ i,e onanyintegrable U q (g) module M. These automorphisms are categorified by certain Rickard Complexes which have played a tremendously important role in a number of recent results, most notably a proof of the Abelian Defect Conjecture. U q (g) itself is not an integrable U q (g)-module, however, the action of τ i,e onintegrablemodulesisverycloselyrelatedto abraidgroupactionbyautomorphismsonthequantum group. These automorphisms satisfy the braid relations and are the key to constructing a PBW type basis for U q (g). In this thesis we categorify these automorphisms and prove a compatibility condition relating them to Rickard complexes. Furthermore, we show that T i,e , our categorified versions of the T i,e , also satisfythebraidrelations. Thiscategorificationhaspotentialimplicationsfor variousongoing projects in categorical representation theory. Contents Chapter 1. Introduction 4 1.1. Categorification 4 1.2. Quantum Groups and their Representations 5 1.3. Categorified quantum groups 5 1.4. Rickard Complexes 6 1.5. Categorifying the Internal Braid Group Action 6 Chapter 2. The quantum group and Lusztig symmetries 8 2.1. The quantum groupU q (g) 8 2.2. (Anti)linear (anti)automorphisms of ˙ U 9 2.3. Lusztig symmetries on integrable ˙ U-modules 10 2.4. Lusztig symmetries on the quantum group 10 Chapter 3. The categorified quantum group 12 3.1. Choice of scalarsQ 12 3.2. Definition of the 2-categoryU Q (g) 12 3.3. Additional relations in U Q 16 3.4. The 2-categories ˙ U Q , Kom( ˙ U Q ), and Com( ˙ U Q ) 17 Chapter 4. Defining the symmetry 2-functorT ′ i,1 21 4.1. T ′ i,1 on objects and 1-morphisms 21 4.2. Definition ofT ′ i,1 on generating 2-morphisms 22 Chapter 5. Categorifying the Braid Group Action ofT ′ i 29 5.1. The Karoubi Envelope of U Q and Thick Calculus 29 5.2. Categorifying the Interaction betweenT ′ i and τ ′ i 32 5.3. Chain homotopy Equivalence i E k 33 5.4. Chain homotopy equivalence i F k 34 5.5. Chain homotopy equivalence i Ej 35 5.6. Chain homotopy equivalence i Fj 44 5.7. Naturality Conditions 51 Chapter 6. The Categorified Braid Group Action 76 6.1. The Braiding of T ′ i 76 6.2. new projects 79 Bibliography 80 3 CHAPTER 1 Introduction 1.1. Categorification Categorifying a mathematical object is a process which involves ’elevating’ the object into a new object that has a more complex mathematical structure than the original object. These ’categorified’ objectsstoreallthe informationcontainedwithin the decategorifiedobject, andbystudying their more elaborate mathematical interactions (which are undetectable on the decategorified level) one hopes to develop a better intuition and deeper understanding of the workings of the original object. However, as we will see, the categorified object can prove to be very interesting in its own right and its study can lead to new insights and deep results about seemingly unrelated topics. We will discuss some familiar examples of categorification to help clarify the somewhat abstract description of the process given above. GivenaCW-complexM wecancomputetheEulercharacteristicχ(M)ofM bytakingthealternat- ing sum of the Betti numbersor equivalently the number of cellsc i in dimensioni. χ(M) = ∞ X i=0 (−1) i c i . The Euler characteristicis a topological invariantthat depends only on the homotopytype of the com- plex and while it is a useful invariant, its calculation does not suggest how the Euler characteristic of two topological spaces M,N are related given a continuous map between them. However, the ho- mology groups H i (M) of a topological space M are a categorification of the Euler characteristic and are a much stronger topological invariant. There is a much richer algebraic structure at play with homology groups due to functoriality. Indeed, a continuous map between the spaces M and N leads to group homomorphisms between the homology groups H i (M) and H i (N). Of course there is also a way to recover the Euler characteristic by ”decategorifying” the homology groups, namely by taking the alternating sum of the dimensions of the homology groups. χ(M) = ∞ X i=0 (−1) i dimH i (M). Thus all the information of the Euler characteristic is stored within its categorification. The decategorification process, in general, is not unique, and will depend on the specific context of the problem. (In fact there can be multiple techniques for decategorying the same categorified object). However, decategorifying by computing the ”Euler characteristic” of a chain complex, i.e. by computing alternating sum of dimensions of vector spaces, is a common technique. The Khovanov homology of a link is a chain complex of graded vector spaces assigned to that link. One can decategorify the Khovanovhomologyof the link by taking the alternating sum of graded dimensions of the vector spaces in each homological degree to arrive at the Jones Polynomial of the link. In other words, Khovanov homology categorifies the Jones polynomial. Khovanov homology is a stronger link invariant than the Jones polynomial, as it detects the unknot, but beyond that, it is functorial. Indeed, a link cobordism between links determines a map between Khovanov homologies of the links. The functorial nature of Khovanov homology has been used to give a proof of the Milnor conjecture. This example illustrates wonderfully the potential power of categorification. 4 1.3. CATEGORIFIED QUANTUM GROUPS 5 1.2. Quantum Groups and their Representations Beginning with a Lie Algebra g, one can construct the Universal Enveloping Algebra U(g). From there one can deform U(g) to produce a non commutative Hopf algebra known as a quantum group U q (g). Beilinson, Lusztig, and MacPherson [1] modified the quantum group U q (g) in order to more appropriately study representations of U q (g) that decompose into weight spaces. It is this algebra, denoted by ˙ U q (g) or ˙ U for short, that will be most important in what follows. A precise algebraic definition of ˙ U q (g) will be given in section 1, but for now we can view it as a free Q(q) algebra with generators E i ,F i , and 1 λ (indexing sets I ∋ i and X ∋ λ will be explained in detail later) modulo a specific set of relations. One such relation is E i F i 1 λ = F i E i 1 λ + [λ i ]1 λ where the quantum integer [λ i ] = q λi −q −λi q−q −1 . Acommonapproachtostudyingcomplexalgebraicobjectsistostudytheirrepresentationsandmuchis knownabouttherepresentationtheoryofquantumgroups. Lusztig[]definesafamilyofautomorphisms of integrable ˙ U q (g)-modules as τ ′ i,e 1 λ = X a,b;a−b=λi (−q) eb F (a) i E (b) i 1 λ = X a,b;a−b=λi (−q) eb E (b) i F (a) i 1 λ , e=±1 (1.1) τ ′′ i,e 1 λ = X a,b;−a+b=λi (−q) eb E (a) i F (b) i 1 λ = X a,b;−a+b=λi (−q) eb F (b) i E (a) i 1 λ . e=±1 (1.2) whereE (a) i =E a i /[a]! andF (a) i =F a i /[a]! Furthermore Lusztig shows that acting on ˙ U by conjugation of these automorphisms defines a family of automorphismsT ′ i,e ,T ′′ i,e of ˙ U itself, despite the fact that ˙ U is not an integrable ˙ U-module. TheT ′ i,e andT ′′ i,e can in fact be uniquely defined as the automorphisms satisfying T ′ i,e (u)τ ′ i,e 1 λ (z) =τ ′ i,e 1 λ (uz), (1.3) T ′′ i,e (u)τ ′′ i,e 1 λ (z) =τ ′′ i,e 1 λ (uz). (1.4) for any u ∈ ˙ U and x ∈ M an integrable ˙ U-module and their action on generators of the quantum group can be explicitly formulated. The automorphismsτ ′ i,e andτ ′ i,e define a braid group action on an integrable ˙ U-module and as a result of this and the defining equations above, it is straightforward to verifythat theT ′ i,e andT ′′ i,e also define a braidgroupaction on ˙ U. We call this the internal braidgroup action. On the decategorified level, ˙ U, the automorphisms τ ′ i,e ,τ ′′ i,e , and the automorphisms T ′ i,e ,T ′′ i,e are well understood. It will, in fact, be the categorification of these automorphisms that will interest us in this paper. 1.3. Categorified quantum groups Studyingtherepresentationsofcategorifiedquantumgroupshasalsoprovedaveryfruitfulendeavor and much is knownabout these so called categoricalrepresentations. When a quantum groupacts on a vector spaceV =⊕ λ V λ the Chevalley generators of the quantum group,E i 1 λ F i 1 λ , act as linear maps on vector spaces. For example,E i 1 λ :V λ →V λ+αi . With a categorical quantum group action, in place of the vector spaces we have additive categoriesV =⊕ λ V λ and the Chevalley generators are replaced by functorsE i 1 λ :V λ →V λ+αi . . Of course the algebraic relations of the quantum group must also be ‘lifted’ to the categorical level too. How does one lift an equation like E i F i 1 λ =F i E i 1 λ +[λ i ]1 λ ? By requiring that there be an isomorphism of functorsE i F i 1 λ ∼ =F i E i 1 λ ⊕ [λi] 1 λ . Decategorification in this setting involves taking the Grothendieck group, K o , of these additive categories. The Grothendieck group is a the free abelian group generated by isomorphism classes of objects modulo the relation [V⊕ 6 1. INTRODUCTION W] = [V]+[W],andinthecaseofcategoricalrepresentations,itrecoverstheirreduciblerepresentations of the quantum group. Motivated in part by many existing instances of these categorical quantum group actions, Aaron Lauda and Mikhail Khovanov successfully categorified the quantum group in 2006 [13, 15, 14]. This entailed the discovery and description of the specific natural transformations that exist between com- posites of the functors E i 1 λ and F i 1 λ mentioned above. Furthermore, they completely described this additional layer of structure using string diagrams, a form of diagrammatic algebra that intuitively expresses all underlying relations between functors. The following example expresses a diagrammatic relation between natural transformations from E i E i 1 λ to itself. • i i − • i i = • i i − • i i = i i The version of the categorified ˙ U that we use in this paper is denoted by U Q . More on the details of this in chapter 3. Independently, Raphael Rouquier developed what appeared to be a slightly different categorification of the quantum group using geometric techniques. Since then, these Khovanov-Lauda- Rouquier algebras, or KLR algebras, have been instrumental in a number of important results in both Algebra and Geometry. ChuangandRouquierusedthesenaturaltransformationstoproveequivalencesofcertaincomplexes in the derived category ofS n modules [12]. These equivalences were fundamental in their proof of the long standing Abelian Defect Conjecture. These same equivalences were fundamental in a proof by Cautis, Kamnitzer, and Licata in which they constructed equivalences between derived categories of coherent sheaves on cotangent bundles of grassmanians [10, 8], thus proving a conjecture of Kawamata in full generality. In a continuing effort to develop a deeper understanding of the existence of categorificationsof link invariants like Khovanov homology, Lauda, Queffelec, and Rose used skew Howe duality to illustrate theconnectionbetweencategorifiedquantumgroupsandcertainfoambased sl n linkhomologytheories [19]. 1.4. Rickard Complexes Figuring prominently in the proofs of each of these results are the so called Rickard complexes, a categorifiedversionof the family of module automorphismsτ ′ i,e 1 λ mentioned above. Because theτ ′ i,e 1 λ are infinite alternating sums of quantum group elements, their categorification requires an appropriate category; one in which we can recover the negative signs when we decategorify. Using Khovanov homology as a guiding example, one might assume that the category of chain complexes Kom(U) be used to categorify theτ ′ i,e 1 λ and the “Euler Characteristic” of the complex be the appropriate method of decategorifying. The deeper structural layer of this categorificationwould appear as the differentials ofthecomplexesintheformofnaturaltransformationsbetweencompositesoffunctorswhichcategorify the quantum group elements E (a) i and F (b) j . While this idea is promising, it turns out that the tensor product of complexes forτ ′ i,e andτ ′−1 i,e is homotopy equivalent, but not equal, to the identity complex. This suggests working in the categoryCom() where the complexes are the same as in Kom() but the morphisms are morphisms of Kom() modulo homotopy. This is indeed the setting in which we work. Given the successful categorificationofτ ′ i,e , it seemslikely that the quantum groupautomorphismsT ′ i,e should also have a categorification since they are so intimately tied to the τ ′ i,e . 1.5. Categorifying the Internal Braid Group Action The goal of this paper is to categorify the braid group action of the family of quantum group automorphisms T ′ i,e and T ′′ i,e . Specifically, we will categorify T ′ i,1 , and note that the categorification of the automorphisms T ′ i,−1 and T ′′ i,e can be achieved using this work along with categorified versions 1.5. CATEGORIFYING THE INTERNAL BRAID GROUP ACTION 7 of certain quantum group symmetries. These symmetries were successfully categorified by Michael Abram in his graduate thesis. The categorification of the braid group action is a 3-step process. In chapter 4 we describe how Lauda, Abram, and the author define the categorified automorphism T ′ i,1 : Com(U Q (g)) → Com(U Q (g)) on all generating 1-morphisms and 2-morphisms in U Q . This definition of T ′ i,1 must not only respect all 2-morphism relations in U Q , but should also recover the definition ofT ′ i,1 on the quantum group when decategorified. Secondly, we categorify the interaction between T ′ i,1 andτ ′ i,1 . Namely, we lift the equation T ′ i,1 (u)τ ′ i,1 1 λ (z) = τ ′ i,1 1 λ (uz) to an isomorphism of functors T ′ i,1 (u1 λ )τ ′ i,1 1 λ ∼ = τ ′ i,1 u1 λ for all u ∈ U Q . We slightly abuse notation here and use the same label for the Rickard complexτ ′ i,e and the ˙ U module automorphism τ ′ i,e . This isomorphism is an equivalence of chain complexes, so for each generating 1-morphismu ofU Q we will calculatei u , the homotopy equivalence betweenT ′ i,1 (u)τ ′ i,1 1 λ andτ ′ i,1 u1 λ . Finally,weshowhowtheseequivalencesarecompatiblewiththehigherstructureofthecategorified quantum group. That is to say, for any 2-morphism f :X →Y between 1-morphisms X,Y we show commutativity of the following diagram. (1.5) τ ′ i,1 y1 λ T ′ i,1 (y)τ ′ i,1 1 λ τ ′ i,1 x1 λ T ′ i,1 (x)τ ′ i,1 1 λ i y i x id τ ⊗f T ′ i,1 (f)⊗id τ Achieving these goals requires many challenging computations with a high level of combinatorial complexity, but the applications of such a categorificationare potentially very broad. A fairly immedi- ate consequence of these computations is the proof that theT ′ i,1 do actually define a braid group action onCom(U Q ). This, in turn, has some consequences that we discuss in the final section of the paper. CHAPTER 2 The quantum group and Lusztig symmetries 2.1. The quantum group U q (g) 2.1.1. Cartandata. AnalgebraicpresentationofaquantumgroupU q (g)canbewritteninterms of the Cartan data of the Lie algebra g. In our categorification of Lusztig’s internal braid group, we restrict ourselves to the simply laced quantum group. The Cartan data for a simply laced Kac-Moody algebra consists of • a free Z-module X (the weight lattice), • for i ∈ I (I is an indexing set) there are elements α i ∈ X (simple roots) and Λ i ∈ X (fundamental weights), • for i∈I an element h i ∈X ∨ = Hom Z (X, Z) (simple coroots), • a bilinear form (·,·) onX. Writeh·,·i: X ∨ ×X → Z for the canonical pairing. These data should satisfy: • (α i ,α i ) = 2 for any i∈I, • hi,λi :=hh i ,λi = (α i ,λ) for i∈I andλ∈X, • (α i ,α j )∈{0,−1} for i,j ∈I with i6=j, • hh j ,Λ i i =δ ij for alli,j∈I. Hence (a ij ) i,j∈I is a symmetrizable generalized Cartan matrix, where a ij = hh i ,α j i = (α i ,α j ). We will sometimes denote the bilinear pairing (α i ,α j ) by i·j and abbreviate hi,λi to λ i . We denote by X + ⊂X the dominant weights which are of the form P i λ i Λ i where λ i ≥ 0. We can associate a Dynkin diagram Γ to a symmetric Cartan data. It is a graph without loops or multiple edges. The vertices of Γ are the elements of the set I and there is an edge from vertex i to vertexj if and only if (α i ,α j ) =−1. If there is no edge between nodes i,k we have (α i ,α k ) = 0. 2.1.2. The simply-laced quantum group. The quantum group U = U q (g) associated to a simply-laced root datum as aboveis the unital associative Q(q)-algebra given by generatorsE i ,F i ,K µ for i∈I and µ ∈X ∨ , subject to the relations: i) K 0 = 1,K µ K µ ′ =K µ +µ ′ for allµ,µ ′ ∈X ∨ , ii) K µ E i =q hµ,α ii E i K µ for all i∈I,µ ∈X ∨ , iii) K µ F i =q −hµ,α ii F i K µ for all i∈I,µ ∈X ∨ , iv) E i F j −F j E i =δ ij Ki−K −1 i q−q −1 , where K i =K αi , v) For all i6=j X a+b=−hi,ji+1 (−1) a E (a) i E j E (b) i = 0 and X a+b=−hi,ji+1 (−1) a F (a) i F j F (b) i = 0, where E (a) i =E a i /[a]!,F (a) i =F a i /[a]!, with [a]! = Q a m=1 q m −q −m q−q −1 . 2.1.3. The integral idempotented form of quantum group. In studying the representation theory of a quantum group, we are primarily interested in the representations that can be decomposed into weight spaces. There is a modified form of the quantum group called ˙ U q g which replaces the 8 2.2. (ANTI)LINEAR (ANTI)AUTOMORPHISMS OF ˙ U 9 unit elements with a family of mutually orthogonal idempotents. We study ˙ U q g because there is an equivalence between the categories of U q (g) representations that have a weight space decomposition and representations of ˙ U. The Q(q)-linear category ˙ U = ˙ U q (g) is defined as follows. The objects of ˙ U are elements ofX. Givenλ,ν ∈X, the hom space is defined as the Q-module ˙ U(λ,ν) :=U/   X µ ∈X ∨ U(K µ −q hµ,λ i )+ X µ ∈X ∨ (K µ −q hµ,ν i )U   . The identity morphism of λ∈X is denoted by 1 λ . The element in ˙ U(λ,µ ) represented by x∈U can be written as 1 µ x1 λ = 1 µ x =x1 λ , where µ −λ =|x|, and E i 1 λ = 1 λ+αi E i , F i 1 λ = 1 λ−αi F i . The composition in ˙ U is induced by multiplication in the algebra, i.e. (1 µ x1 ν )(1 ν y1 λ ) = 1 µ xy1 λ for x,y∈U, λ,µ,ν ∈X, which is zero unless|x| =µ −ν, |y| =ν−λ. Let A = Z[q,q −1 ]. The algebra ˙ U admits an integral form A ˙ U defined as the Z[q,q −1 ]-linear sub-lattice of ˙ U spanned by products of divided powersE (a) i 1 λ andF (a) i 1 λ . 2.2. (Anti)linear (anti)automorphisms of ˙ U The families of automorphisms T ′ i,e and T ′′ i,e are related to each other. There is a set of Z[q,q −1 ] automorphisms (involutions) of ˙ U which act on T ′ i,e and T ′′ i,e by conjugation. The inverses of each of theT ′ i,e can be generated by this action. These involutions were categorified by Abram in his graduate thesis and the action of these categorified versions allow one to defineT ′ i,−1 andT ′′ i,e in terms ofT ′ t,1 at the catefgorified level. We define the involutions here. Let¯be the Q-linear involution of Q(q) which maps q to q −1 . • The Q(q)-linear algebra antiinvolutionσ: U→U is given by σ(E i ) =E i , σ(F i ) =F i , σ(K i ) =K −1 i , σ(fx) =fσ(x), forf ∈ Q(q) andx∈U, σ(xy) =σ(y)σ(x), forx,y∈U. • The Q(q)-linear algebra involutionω: U→U is given by ω(E i ) =F i , ω(F i ) =E i , ω(K i ) =K −1 i , ω(fx) =fω(x), forf ∈ Q(q) andx∈U, ω(xy) =ω(x)ω(y), forx,y∈U. • The Q(q)-antilinear algebra involutionψ: U→U is given by ψ(E i ) =E i , ψ(F i ) =F i , ψ(K i ) =K −1 i , ψ(fx) = ¯ fψ(x), forf ∈ Q(q) andx∈U, ψ(xy) =ψ(x)ψ(y), forx,y∈U. The (anti)linear (anti)involutions σ,ω, and ψ pairwise commute and naturally extend to 1 λ ′ ˙ U1 λ if we set σ(1 λ ) = 1 −λ , ω(1 λ ) = 1 −λ , ψ(1 λ ) = 1 λ . We can extend these symmetries to ˙ U and A ˙ U by taking direct sums of the induced maps on each summand 1 λ ′ ˙ U1 λ . 10 2. THE QUANTUM GROUP AND LUSZTIG SYMMETRIES 2.3. Lusztig symmetries on integrable ˙ U-modules Fore=±1 andM an integrable ˙ U-module, Lusztig [20, 5.2.1] defines linear mapsτ ′ i,e ,τ ′′ i,e :M → M by τ ′ i,e (m) = X a,b,c;a−b+c=λi (−1) b q e(−ac+b) F (a) i E (b) i F (c) i m, (2.1) τ ′′ i,e (m) = X a,b,c;−a+b−c=λi (−1) b q e(−ac+b) E (a) i F (b) i E (c) i m, (2.2) for m∈M(λ). Furthermore, these maps are automorphisms of M with inverses given by τ ′ i,e τ ′′ i,−e =τ ′′ i,−e τ ′ i,e = Id: M →M. (2.3) These maps satisfy the braid relations on any integrable module M [20, Theorem 39.4.3]. τ ′ i,e τ ′ j,e τ ′ i,e =τ ′ j,e τ ′ i,e τ ′ j,e τ ′′ i,e τ ′′ j,e τ ′′ i,e =τ ′′ j,e τ ′′ i,e τ ′′ j,e if i·j =−1 τ ′ i,e τ ′ j,e =τ ′ j,e τ ′ i,e τ ′′ i,e τ ′′ j,e =τ ′′ j,e τ ′′ i,e if i·j = 0 By restricting the action of these automorphisms to individual weight spaces M(λ) of integrable modules M, we can write these automorphisms in a much simpler form. τ ′ i,e 1 λ = X a,b;a−b=λi (−q) eb F (a) i E (b) i 1 λ = X a,b;a−b=λi (−q) eb E (b) i F (a) i 1 λ , τ ′′ i,e 1 λ = X a,b;−a+b=λi (−q) eb E (a) i F (b) i 1 λ = X a,b;−a+b=λi (−q) eb F (b) i E (a) i 1 λ . (2.4) This form results from some algebraic manipulations involving the symmetries ω,ψ, and [11, Lemma 6.1.1]. 2.4. Lusztig symmetries on the quantum group For each i ∈ I and e = ±1, Lusztig defines algebra automorphisms T ′ i,e and T ′′ i,e of ˙ U = ˙ U q (g) defined in [20] as follows: T ′ i,e (1 λ ) = 1 si(λ) T ′ i,e (E ℓ 1 λ ) =    −q −e(2+λi) F i 1 si(λ) if i=ℓ E ℓ E i 1 si(λ) −q e E i E ℓ 1 si(λ) if i·ℓ =−1 E ℓ 1 si(λ) if i·ℓ = 0 (2.5) T ′ i,e (F ℓ 1 λ ) =    −q e(λi) E i 1 si(λ) if i=ℓ F i F ℓ 1 si(λ) −q −e F ℓ F i 1 si(λ) if i·ℓ =−1 F ℓ 1 si(λ) if i·ℓ = 0 T ′′ i,e (1 λ ) = 1 si(λ) T ′′ i,e (E ℓ 1 λ ) =    −q −e(λi) F i 1 si(λ) if i =ℓ E i E ℓ 1 si(λ) −q −e E ℓ E i 1 si(λ) if i·ℓ =−1 E ℓ 1 si(λ) if i·ℓ = 0 (2.6) T ′′ i,e (F ℓ 1 λ ) =    −q e(λi−2) E i 1 si(λ) if i =ℓ F ℓ F i 1 si(λ) −q e F i F ℓ 1 si(λ) if i·ℓ =−1 F ℓ 1 si(λ) if i·ℓ = 0 2.4. LUSZTIG SYMMETRIES ON THE QUANTUM GROUP 11 where s i is the element of the Weyl group of g corresponding to the simple root α i . Thus, s i (λ) = λ−hi,λiα i =λ−λ i α i . The Lusztig operators are invertible. In particular, the inverse of T ′ i,e is T ′′ i,−e [20, 41.1.1]. The automorphismsT ′ i,e andT ′′ i,e are related by σT ′ i,e σ =T ′′ i,−e ωT ′ i,e ω =T ′′ i,e , (2.7) ψT ′ i,e ψ =T ′ i,−e ψT ′′ i,e ψ =T ′′ i,−e . (2.8) As a consequence, we see that the Lusztig symmetry operators are invariant under conjugation by the triple composite σωψ, that is, σωψT ′ i,e ψωσ =T ′ i,e and σωψT ′′ i,e ψωσ =T ′′ i,e . In what follows, we focus on the automorphism T ′ i,1 . Similar statements can be made for the automorphisms T ′′ i,−1 , T ′′ i,1 , and T ′ i,−1 by applying the corresponding symmetry operators σ, ω, or ψ. When the context is clear, we will abbreviate T ′ i,1 by T ′ i . Lusztig shows in [20, 39.2.4 and 39.2.5] that the formulas (2.5) define an action of the braid group of type g on ˙ U. In particular, the automorphisms satisfy T ′ i T ′ ℓ T ′ i =T ′ ℓ T ′ i T ′ ℓ if i·ℓ =−1, (2.9) T ′ i T ′ ℓ =T ′ ℓ T ′ i if i·ℓ = 0. (2.10) 2.4.1. Compatibility of braid group actions. For any integrable ˙ U-module M, any z ∈ M, and u ∈ ˙ U, the braid group action on modules and the braid group action on ˙ U are related [20, Proposition 37.1.2] by the equations T ′ i,e (u)τ ′ i,e 1 λ (z) =τ ′ i,e 1 λ (uz), T ′′ i,e (u)τ ′′ i,e 1 λ (z) =τ ′′ i,e 1 λ (uz). (2.11) Indeed, the algebra automorphisms T ′ i,e and T ′′ i,e are uniquely determined by these equations. Put another way, there is an equality of maps between endomorphisms ofM T ′ i,e (u1 λ )τ ′ i,e 1 λ =τ ′ i,e u1 λ T ′′ i,e (u1 λ )τ ′′ i,e 1 λ =τ ′′ i,e u1 λ (2.12) for allu1 λ ∈ ˙ U. To prove this, it suffices to consider u =E j 1 λ oru =F j 1 λ . CHAPTER 3 The categorified quantum group Here we describe a categorification of U(g) mainly following [2] and [9, 14]. For an elementary introduction to the categorification of sl 2 see [18]. 3.1. Choice of scalars Q Let k be an field, not necessarily algebraically closed, or characteristic zero. Definition 3.1.1. Associatedto asymmetricCartandatum definea choice of scalarsQconsisting of: • {t ij | for alli,j ∈I}, such that • t ii = 1 for all i∈I and t ij ∈k × for i6=j, • t ij =t ji when a ij = 0. We say that a choice of scalarsQ is integral if t ij =±1 for alli,j ∈I. The choice of scalarsQ controlsthe form of the KLR algebraR Q that governsthe upwardoriented strands. The 2-categoryU Q (g) are governed by the products v ij =t −1 ij t ji taken over all pairs i,j ∈I. Theproductsv ij canbethoughtofasak × -valued1-cocycleonthegraphΓassociatedtothesymmetric Cartan data; we call two choices cohomologous if these 1-cocycles are in the same cohomology class. Obviously if Γ is a tree, in particular, a Dynkin diagram, then all choices of scalars are cohomologous. Definition 3.1.2. Associated to a Cartan datum we define bubble parameters C to be a set consisting of • c i,λ ∈k × for i∈I andλ∈X. These data are said to be compatible with the scalars Q if (3.1) c i,λ+αj /c i,λ =t ij . Such a compatible choiceof scalarscanbe chosenfor anyt ij byfixing an arbitrarychoiceofc i,λ for a fixed coset representative in every coset of the root lattice in the weight lattice, and then extending to the rest of the coset using the compatibility condition. For any choice of bubble parameters compatible with the choice of scalars Q the values along an sl 2 -string remain constant since c i,λ+αi =t ii c i,λ =c i,λ , so that for alln∈ Z we have c i,λ =c i,λ+nαi . 3.2. Definition of the 2-category U Q (g) When mentally organizing the multilayered structure of a 2-category, it may be helpful to keep in mind the category of categoriesCat, in which there are three layers of structure. Categories act as the objects, functors (between categories) act as morphisms between objects, and natural transformations 12 3.2. DEFINITION OF THE 2-CATEGORY UQ(g) 13 act as morphisms between functors. Similarly, 2-categories have objects, 1-morphisms (morphisms be- tween objects), and 2-morphisms (morphisms between compositions of 1-morphisms).The categorified quantum group U Q (g) is a graded linear 2-category by which we mean it is a category enriched in graded linear categories. Thus, the 2-morphisms have the additional structure of vector spaces, and we admit all finitary products of 1-morphisms. A graded linear category is a category equipped with an auto-equivalence h1i. We denote by hti the auto-equivalence obtained by applying h1i t times and by h−ti the auto-equivalence obtained by applying the inverse of h1i t times. Thus, given any object x∈ Ob(C), there exists an object xhti∈ Ob(C) for all t∈ Z. Likewise, given any morphism f: x→y in C, there is a morphism fhti: xhti→yhti. Composition of morphisms preserves the grading. Given a fixed choice of scalars Q and bubble parameters C we recall the definition from [2] of the following 2-category. Definition 3.2.1. The 2-categoryU Q :=U cyc Q (g) is the graded linear 2-category consisting of: • objects λ for λ∈X. • 1-morphisms are formal direct sums of (shifts of) compositions of 1 l , 1 λ+αi E i = 1 λ+αi E i 1 λ =E i 1 λ , and 1 λ−αi F i = 1 λ−αi F i 1 λ =F i 1 λ for i ∈ I and λ ∈ X. In particular, any morphism can be written as a finite formal sum of symbolsE i 1 λ hti where i = (±i 1 ,...,±i m ) is a signed sequence of simple roots,t is a grading shift, E +i 1 λ :=E i 1 λ andE −i 1 λ :=F i 1 λ , andE i 1 λ hti :=E ±i1 ...E ±im 1 λ hti. • 2-morphismsarefree Z-modules spanned by compositionsof (decorated) tangle-likediagrams illustrated below. • i λ λ+αi : E i 1 λ →E i 1 λ h2i • i λ λ−αi : F i 1 λ →F i 1 λ h2i i j λ : E i E j 1 λ →E j E i 1 λ h−i·ji i j λ : F i F j 1 λ →F j F i 1 λ h−i·ji i j λ : F i E j 1 λ →E j F i 1 λ i j λ : E i F j 1 λ →F j E i 1 λ i λ : 1 λ →F i E i 1 λ h1+λ i i i λ : 1 λ →E i F i 1 λ h1−λ i i i λ : F i E i 1 λ → 1 λ h1+λ i i i λ : E i F i 1 λ → 1 λ h1−λ i i Here we follow the grading conventions in [9] and [19] which are opposite to those from [14]. We readdiagramsfromrightto leftandbottomto top. Theidentity2-morphismofthe1-morphismE i 1 λ is representedby an upwardoriented line labeled byi and the identity 2-morphismofF i 1 λ is represented by a downward such line. Sometimes we will color the strands to help visually distinguish their labels. The 2-morphisms satisfy the following 10 relations: (1) Right and left adjunction relations λ λ+α i = λ λ+α i λ+α i λ = λ+α i λ 14 3. THE CATEGORIFIED QUANTUM GROUP λ λ+α i = λ λ+α i λ+α i λ = λ+α i λ (2) Dot cyclicity relations λ+α i λ • = • λ λ+α i = λ+α i λ • (3) Crossing cyclicity relations i j λ = λ j i j i = λ i j i j We can then define a sideways crossing in terms of an upwards crossing. The second equality (that we could have defined a sideways crossing in terms of a downwards crossing) will follow from the crossing cyclcity relations and the right and left adjunction relations. j i λ = λ i j i j = λ j i j i i j λ = λ j i j i = λ i j i j The next three relations imply that theE’s (andF ′ s) carry an action of the KLR algebra associated to Q. The KLR algebra is generated by the upwards line, upwards dot, and upwards crossing diagrams. Its Karoubi envelope categorifies A ˙ U + . (4) R2 relations λ i j =                            0 if i =j, t ij i j if i·j = 0, t ij • i j + t ji • i j if i·j =−1, 3.2. DEFINITION OF THE 2-CATEGORY UQ(g) 15 (5) Dot slide relations • i j − • i j = • i j − • i j =      i i if i =j 0 if i6=j (6) R3 relations (3.2) λ i j k − λ i j k =                t ij i j i if i=k and i·j =−1 0 if i6=k or i·j 6=−1 (7) Bubble relations i • λi−1+m λ = 0 for m< 0 i • −λi−1+m λ = 0 for m< 0 i • λi−1+0 λ =c i,λ Id 1 λ i • −λi−1+0 λ =c −1 i,λ Id 1 λ We introduce formal symbols called fake bubbles. These are positive degree endomor- phisms of 1 λ that carry a formal label by a negative number of dots. For λ i < 0 i • λi−1+j λ =          −c i,λ X a+b=j b≥1 • λi−1+a • −λi−1+b λ if 0<j <−λ i +1 0 if j ≤ 0. For λ i > 0 i • −λi−1+j λ =          −c −1 i,λ X a+b=j a≥1 • λi−1+a • −λi−1+b λ if 0<j <λ i +1 0 if j ≤ 0. (8) Extended sl 2 relations Note that in [9] some additional curl relations were provided that can be derived from the relations below together with the crossing cyclicty relations [2, 3.2]. We employ the convention that all summations are increasing, so that P α f=0 is zero if α< 0. i i λ = − λ i i + X a+b+c =λi−1 λ • c • a i • −λi−1+b i i 16 3. THE CATEGORIFIED QUANTUM GROUP i i λ = − λ i i + X a+b+c =−λi−1 • c • a i • λi−1+b i i λ (9) Mixed EF relations λ i j = λ i j λ i j = λ i j Remark 3.2.2. If two choices of scalars Q and Q ′ are cohomologous, then rescaling dots and crossingsinduces an isomorphismU Q (g) ∼ =U Q ′(g). This is shown for the KLR algebra in [15], and this suffices by the presentation given by Brundan in [5]. Remark 3.2.3. It is often helpful to work with a reduced presentation forU Q where we restrict to the following generating 2-morphisms: • i λ λ+αi : E i 1 λ →E i 1 λ h2i i j λ :E i E j 1 λ →E j E i 1 λ h−i·ji i λ : 1 λ →F i E i 1 λ h1+λ i i i λ : 1 λ →E i F i 1 λ h1−λ i i i λ : F i E i 1 λ → 1 λ h1+λ i i i λ : E i F i 1 λ → 1 λ h1−λ i i Thedownwardpointing dot2-morphism, alongwith sidewaysand downwardcrossingsarethen defined using the cyclicity conditions. This presentation can be further simplified by requiring a smaller set of axiomsandthe localizationof certainmaps [6, 23]. However,this further reducedtheoryis not helpful in this article since showing that the required maps are invertible requires verifying the axioms in U Q . The further reduced presentation from [6, 23] is helpful for checking biadjointness and cyclicity, but for our purposes this in not needed. 3.3. Additional relations in U Q In this section we collect several relations that will be used in latter sections. 3.3.1. Curl relations. The following relations allow dotted curls to be reduced to simpler dia- grams. (3.3) λ • m i = − X f1+f2 =m−hi,λi λ i i • hi,λi−1+f2 • f1 λ • m i = X g1+g2 =m+hi,λi i λ i • −hi,λi−1+g2 •g1 See [4] for a derivation of these equations. 3.3.2. Infinite Grassmannian relations. The infinite Grassmannian relations are obtained by comparing powers oft in the expression below.   i • −λi−1 λ + i • −λi−1+1 λ t+···+ i • −λi−1+α λ t α +···     i • λi−1 λ + i • λi−1+1 λ t+···+ i • λi−1+α λ t α +···   = Id 1 λ 3.4. THE 2-CATEGORIES ˙ UQ, Kom( ˙ UQ), AND Com( ˙ UQ) 17 For lowpowersoftthis is just the definition of fakebubbles in terms ofrealbubbles. For higher powers oft this relation can be proven from the others using curl relations and the extended sl 2 -relations. 3.3.3. Triple intersections. Using cyclicity and (3.2) one can show that for all i,j,k ∈ I the equation (3.4) λ i j k = λ i j k holds in U Q (g) unless i=j =k. Similarly, it is not hard to show that if (α i ,α j ) =−1, then (3.5) j i i − i i j = t ij i i j (3.6) i i j − j i i = t ij i i j also hold in U Q . 3.4. The 2-categories ˙ U Q , Kom( ˙ U Q ), and Com( ˙ U Q ) 3.4.1. Additive categories, homotopy categories, and Karoubi envelopes. Given an ad- ditive categoryM we write Kom(M) for the category of bounded complexes inM. An object (X,d) ofKom(M) is a collection of objects X i ofM together with maps d i (3.7) ... di−2 X i−1 di−1 X i di X i+1 di+1 ... such that d i+1 d i = 0 and only finitely many objects are nonzero. A morphism f: (X,d) → (Y,d) in Kom(M) is a collection of morphisms f i : X i →Y i such that (3.8) ... di−2 X i−1 di−1 fi−1 X i di fi X i+1 di+1 fi+1 ... ... di−2 Y i−1 di−1 Y i di Y i+1 di+1 ... commutes. For a pair of morphismsf,g: (X,d)→ (Y,d) inKom(M), we say thatf is homotopic tog if there exist morphismsh i : X i →Y i−1 such thatf i −g i =h i+1 d i +d i−1 h i for alli. A morphism of complexes is said to be null-homotopic if it is homotopic to the zero map. Definition 3.4.1. The homotopy category Com(M) has the same objects as Kom(M) and its morphisms are morphisms in Kom(M) modulo null-homotopic morphisms. IfM is a monoidal additive category then bothCom(M) andKom(M) are also monoidal. Given two bounded complexes (X,d X ) and (Y,d Y ) define their tensor product (XY,d) as the complex with (3.9) (XY) i = M r+s=i X r Y s , d i := X r (d X ) r 1 Y i−r +(−1) r 1 X r(d Y ) i−r 18 3. THE CATEGORIFIED QUANTUM GROUP wherewedenotethetensorproductofobjectsandmorphismsinMbyjuxtaposition. Givenchainmaps f: (X,d X )→ (X ′ ,d X ′) andg: (Y,d Y )→ (Y ′ ,d Y ′) define the tensor productfg: (XY,d)→ (X ′ Y ′ ,d ′ ) of chain maps by setting (3.10) f i = M r+s=i f r g s . It is straight-forward to check that the tensor product of homotopic complexes will be homotopic. Definition 3.4.2. Given an object ofKom(Kom(M)) we can form a bicomplex; that is, a collec- tionX r,s of objects of M together with maps d h r,s : X r,s →X r+1,s , d v r,s : X r,s →X r,s+1 suchthatd h ◦d h =d v ◦d v =d v d h +d h d v = 0. We then define the total complex Tot(X) of a bicomplex by Tot(X) i := M r+s=i X r,s with differentiald r,s =d h r,s +(−1) r d v r,s . The total complex construction extends to an additive functor Tot: Kom(Kom(M))→Kom(M). Constructions and concepts from homological algebra appear often in categorification. 3.4.2. Gaussian Elimination. In chapter 5, we construct homotopy equivalences between com- plexes of 1-morphisms of ˙ U . The process relies heavily on Gaussian Elimination, a technique in homological algebra used for simplifying a chain complex into a homotopy equivalent complex. We explain some details of this mechanism here. Given maps A :X →W, B :Y →W, C :X →Z, and D :Y →Z with D an isomorphism, there is a homotopy equivalence of complexes (3.11) U X⊕Y W ⊕Z [ A B C D ] V [ ux uy ] [wv zv] X W U V [ A−BD −1 C ] 1X −D −1 C h 1W 0 i [ux] [wv] [1U ] [1V ] This can be verified by checking that the maps 0 0 0 −D −1 :W ⊕Z →X⊕Y and [0] :W →X satisfy the necessary homotopy conditions in the following commutative diagram: 3.4. THE 2-CATEGORIES ˙ UQ, Kom( ˙ UQ), AND Com( ˙ UQ) 19 (3.12) U U U U X⊕Y W ⊕Z V V V V [ A B C D ] X W [ A−BD −1 C ] 1X −D −1 C h 1W 0 i [1U ] [1U ] [1U ] [1V ] [1V ] [1V ] X⊕Y W ⊕Z [ A B C D ] X W [ A−BD −1 C ] [1X 0] [1W −BD −1 ] 1X −D −1 C h 1W 0 i [ ux uy ] [ux] [ ux uy ] [ux] [wv zv] [wv] [wv zv] [wv] We should note that the chain map shown above can be multiplied by any invertible scalare c (and it’s inverseequivalencemultiplied bye c −1 )andtheresultingchainmapwillstillbeahomotopyequivalence. An important variation of this homotopy equivalence occurs when the matrix of maps [ A B C D ] is upper block triangular with C = 0. In this case, there is a homotopy equivalence of complexes (3.13) U X⊕Y W ⊕Z [ A B 0 D ] V [ ux uy ] [wv zv] X W U V [A] h 1X 0 i h 1W 0 i [ux] [wv] [1U ] [1V ] This case appears often when proving the homotopy equivalence betweenτ ′ i,1 E j 1 λ and T ′ i,1 (E j )τ ′ i,1 1 λ . 3.4.3. Karoubian envelope of U. We give some preliminary information about the Karoubi envelope Kar(U) of U Q (g) and we will discuss the algebraic construction in more detail in chapter 5. For now we note that Kar(U) is an enlargement of U into which there is an inclusion of U. The Karoubi completion and passage to the (homotopy) category of complexes commute with each other. Proposition 3.4.3 (Proposition 3.6 [3]). For any additive category M there exists a canonical equivalence (3.14) Kom(Kar(M)) ∼ =Kar(Kom(M)). Proposition 3.4.4 (Corollary 2.12 [7] or Proposition 3.7[3]). For any additive k-linear category M with finite dimensional hom spaces there exists a canonical equivalence (3.15) Com(Kar(M)) ∼ =Kar(Com(M)). Definition 3.4.5. Define the additive 2-category ˙ U Q to have the same objects asU Q and hom ad- ditivecategoriesgivenby ˙ U Q (λ,λ ′ ) =Kar(U Q (λ,λ ′ )). The fully-faithful additivefunctorsU Q (λ,λ ′ )→ 20 3. THE CATEGORIFIED QUANTUM GROUP ˙ U Q (λ,λ ′ ) combine to form an additive 2-functorU Q → ˙ U Q universal with respect to splitting idempo- tents in the hom categories ˙ U Q (λ,λ ′ ). The composition functor ˙ U Q (λ ′ ,λ ′′ )× ˙ U Q (λ,λ ′ )→ ˙ U Q (λ,λ ′′ ) is induced by the universal property of the Karoubi envelope from the composition functor for U Q . The 2-category ˙ U Q has graded 2-homs given by (3.16) HOM ˙ UQ (x,y) := M t∈ Z Hom ˙ UQ (x,yhti). The significance of the 2-categoryU Q (g) is given by the following theorem. Theorem 3.4.6. ([17, 14, 24]) Working with k-linear combinations of diagrams, rather than Z-linear, so that U Q (g) is a graded k-linear 2-category, there is an isomorphism γ: A ˙ U −→ K 0 ( ˙ U Q (g)) (3.17) where K 0 ( ˙ U) is the split Grothendieck ring of ˙ U Q . This isomorphism maps the semilinear form h·,·i on A ˙ U to the graded Hom form HOM ˙ UQ (·,·). For sl 2 this theorem follows over Z by the results in [16]. 3.4.4. Karoubian envelopes of Kom(U) and Com(U). Note that the 2-homs U Q (x,yhti) are finite-dimensional k-vector space for eacht∈ Z. Definition 3.4.7. Define Kom(U Q ) to be the additive 2-category with objects λ ∈ X and additive hom categories Kom(U Q )(λ,λ ′ ) := Kom(U Q (λ,λ ′ )). The additive composition functor Kom(U Q (λ ′ ,λ ′′ ))×Kom(U Q (λ,λ ′ ))→Kom(U Q (λ,λ ′′ )) is given by the tensor product of complexes using the additive composition functor onU Q to tensor 1-morphisms via composition. Definition 3.4.8. Define Com(U Q ) to be the additive 2-category with the same objects and 1- morphisms as Kom(U Q ) and 2-morphisms given by identifying homotopy equivalent 2-morphisms in Kom(U Q ). Recall that ˙ U Q =Kar(U Q ). By Propositions 3.4.3 and 3.4.4 there are equivalences (3.18) Kar(Kom(U Q )) ∼ =Kom( ˙ U Q ), Kar(Com(U Q )) ∼ =Com( ˙ U Q ). The2-categoriesweconsiderfit into thefollowing tablewherethe horizontalarrowsdenotepassage to the Karoubian envelope and vertical arrows stand for passage to complexes and modding out by null-homotopic maps. (3.19) U Q ˙ U =Kar(U Q ) Kom(U Q ) Kom( ˙ U Q ) ∼ =Kar(Kom(U)) Com(U Q ) Com( ˙ U Q ) ∼ =Kar(Com(U)) Theorem 3.4.6 and the main result of [22] imply that (3.20) K 0 (Kar(Kom(U Q ))) ∼ =K 0 (Kom(Kar(U Q ))) =K 0 (Kar(U Q )) ∼ =K 0 ( ˙ U Q ) ∼ = A ˙ U. Hence, we can view the Karoubi envelope of the category of complexesKom(U Q ) as a categorification of the integral idempotent form A ˙ U of the quantum group. CHAPTER 4 Defining the symmetry 2-functor T ′ i,1 In thischapter weexplicitlydefine additivek-linear2-functorsT ′ i,1 : U Q →Com(U Q ) foreachi∈I. We first define these 2-functors on the generators of U Q and extend 2-functorially. In particular, 1- morphismsofU Q aremappedtocomplexesof1-morphismsandweextendtocompositesof1-morphisms by taking tensor products of complexes. Similarly, 2-morphisms ofU Q are mapped to chain maps and we extend to vertical composites of 2-morphisms by taking composites of chain maps and to horizontal composites of 2-morphisms by taking tensor products of chain maps. We also extendT ′ i,1 additively to direct sums. Aaron Lauda, Mike Abram, and the aurthor proved that T ′ i,1 is well-defined by showing that it preserves all 10 defining relations on 2-morphisms of U up to chain homotopy. We omit the explicit computations of the various chain homotopies involved in this proof, but direct the interested reader to [?] for a full detailed description. Let g be a symmetric Kac-Moody algebra. Then the data given below T ′ i,+1 :Com( ˙ U Q (g))→Com( ˙ U Q (g)) defines a 2-functor that induces the map [T ′ i,+1 ] =T ′ i,+1 : ˙ U Z (g)→ ˙ U Z (g) on K 0 ( ˙ U Q (g)) ∼ = ˙ U Z (g). The other versions of Lusztig’s categorified operators can then be computed using the symmetries in section 2.2. Namely, T ′′ i,−1 := σT ′ i,1 σ, T ′′ i,1 := ωT ′ i,1 ω, T ′ i,−1 := ψT ′ i,1 ψ, For every i∈I we define an additive k-linear 2-functor T ′ i,1 :U Q →Com(U Q ). We abbreviate T ′ i,1 by T ′ i when the context is clear. 4.1. T ′ i,1 on objects and 1-morphisms On objects the 2-functorT ′ i,1 is defined by T ′ i (λ) =s i (λ) Recall that s i is the Weyl group element defined by s i (λ) =λ−λ i α i . On generating 1-morphismsT ′ i,1 is given by complexes 21 22 4. DEFINING THE SYMMETRY 2-FUNCTOR T ′ i,1 T ′ i (1 λ ) = 0 0 ♣ 1 si(λ) T ′ i (E ℓ 1 λ ) =                  0 ♣ 0 F i 1 si(λ) h−2−λ i i if i =ℓ ♣E ℓ E i 1 si(λ) E i E ℓ 1 si(λ) h1i ℓ i if i·ℓ =−1 0 0 ♣E ℓ 1 si(λ) if i·ℓ = 0 T ′ i (F ℓ 1 λ ) =                  ♣ 0 0 E i 1 si(λ) hλ i i if i =ℓ F ℓ F i 1 si(λ) h−1i ♣F i F ℓ 1 si(λ) ℓ i if i·ℓ =−1 0 0 ♣F ℓ 1 si(λ) if i·ℓ = 0. Since our complexes have at most two nonzero components, it is trivial that the square of the differential is zero. We extend the definition to compositionsof 1-morphismsby taking tensor products of complexes. (Note that the clubsuit denotes homological degree zero.) 4.2. Definition of T ′ i,1 on generating 2-morphisms As a convenience we will denotei-labelled strands in red,j-labelled strands in black (wherei·j = −1), and k-labelled strands in green (where i·k = 0) unless stated otherwise. Also, we often omit labelling the weights i (λ) in the far right region of the diagrams. Definition of T ′ i,1 on upwards dot 2-morphisms. T ′ i • i λ λ+α i := F i 1 si(λ) h−λ i i F i 1 si(λ) h−2−λ i i • i T ′ i • k λ λ+α k := ♣E k 1 si(λ) h2i ♣E k 1 si(λ) • k 4.2. DEFINITION OF T ′ i,1 ON GENERATING 2-MORPHISMS 23 T ′ i • j λ λ+α j := ♣E j E i 1 si(λ) h2i ♣E j E i 1 si(λ) • j i E i E j 1 si(λ) h3i E i E j 1 si(λ) h1i i • j j i j i Definition of T ′ i,1 on upwards crossing 2-morphisms. T ′ i i i λ := F i F i 1 si(λ) h−8−2λ i i F i F i 1 si(λ) h−6−2λ i i − i i T ′ i k k ′ λ := ♣E k ′E k 1 si(λ) h−k·k ′ i ♣E k E k ′1 si(λ) k k ′ T ′ i i k λ := E k F i 1 si(λ) h−2−λ i i F i E k 1 si(λ) h−2−λ i i i k t ki T ′ i k i λ := F i E k 1 si(λ) h−2−λ i i E k F i 1 si(λ) h−2−λ i i k i T ′ i i j λ := E j E i F i 1 si(λ) h−1−λ i i F i E j E i 1 si(λ) h−1−λ i i i j i ♣ E i E j F i 1 si(λ) h−λ i i ♣ F i E i E j 1 si(λ) h−λ i i − i i j i j i i j i − 24 4. DEFINING THE SYMMETRY 2-FUNCTOR T ′ i,1 T ′ i j i λ := F i E j E i 1 si(λ) h−λ i i E j E i F i 1 si(λ) h−2−λ i i i i j • t ij −t ij i i j • ♣F i E i E j 1 si(λ) h1−λ i i ♣E i E j F i 1 si(λ) h−1−λ i i j i i t ij • −t ij j i i • i j i i j i − T ′ i j k λ := ♣E k E j E i 1 si(λ) h−j·ki ♣E j E i E k 1 si(λ) t −1 ki i k j E k E i E j 1 si(λ) h1−j·ki E i E j E k 1 si(λ) h1i t −1 ki j k i k j i k j i T ′ i k j λ := ♣E j E i E k 1 si(λ) h−k·ji ♣E k E j E i 1 si(λ) k j i E i E j E k 1 si(λ) h1−k·ji E k E i E j 1 si(λ) h1i k i j k j i k j i The most complicated crossing is colored by nodes j and j ′ with j·i=j ′ ·i =−1. T ′ i j j ′ λ := 4.2. DEFINITION OF T ′ i,1 ON GENERATING 2-MORPHISMS 25 ♣E j E i E j ′E i 1 si(λ) E i E j E j ′E i 1 si(λ) h1i E j E i E i E j ′1 si(λ) h1i E i E j E i E j ′1 si(λ) h2i j i i j ′ j ′ i i j j ′ i i − j j i j ′ i ♣E j ′E i E j E i 1 si(λ) h−j·j ′ i E i E j ′E j E i 1 si(λ) h1−j·j ′ i E j ′E i E i E j 1 si(λ) h1−j·j ′ i E i E j ′E i E j 1 si(λ) h2−j·j ′ i i j ′ i j t −1 ij j i j ′ i −t −1 ij i i j j −δ jj ′v ij j j ′ i i t −1 ij t ij ′ t −1 ij j j ′ i i t −1 ij i i j ′ j 4.2.1. Definition of T ′ i,1 on cap and cup 2-morphisms. T ′ i i λ ! := ♣ 1 si(λ) h1−λ i i ♣ F i E i 1 si(λ) i s i (λ) c i,λ T ′ i i λ ! := ♣ 1 si(λ) ♣E i F i 1 si(λ) h1+λ i i i s i (λ) c −1 i,λ T ′ i i λ ! := ♣ 1 si(λ) ♣ F i E i 1 si(λ) h1−λ i i i s i (λ) c i,λ T ′ i i λ ! := ♣ 1 si(λ) h1+λ i i ♣E i F i 1 si(λ) i s i (λ) c −1 i,λ The maps have the correct degree since 1±hi,s i (λ)i = 1±hi,λ−λ i α i i = 1±λ i ∓2λ i = 1∓λ i . 26 4. DEFINING THE SYMMETRY 2-FUNCTOR T ′ i,1 For i·k = 0 then T ′ i k λ ! := ♣ 1 si(λ) h1−λ k i ♣ E k F k 1 si(λ) k t λi ki s i (λ) T ′ i k λ ! := ♣ 1 si(λ) ♣F k E k 1 si(λ) h1+λ k i k t −λi ki s i (λ) T ′ i k λ ! := ♣ 1 si(λ) ♣E k F k 1 si(λ) h1−λ k i k s i (λ) T ′ i k λ ! := ♣ 1 si(λ) h1+λ k i ♣F k E k 1 si(λ) k s i (λ) The maps have the correct degree since 1±hk,s i (λ)i = 1±hk,λ−λ i α i i = 1±λ k . For i·j =−1 then T ′ i   j λ   := E j E i F j F i 1 si(λ) hλ j −2i ♣E i E j F j F i 1 si(λ) hλ j −1i ♣E j E i F i F j 1 si(λ) hλ j −1i E i E j F i F j 1 si(λ) hλ j i − 0 ♣ 1 si(λ) 0 −(−1) λj c −1 j,λ (−1) λj c −1 j,λ 4.2. DEFINITION OF T ′ i,1 ON GENERATING 2-MORPHISMS 27 T ′ i j λ := F j F i E j E i 1 si(λ) hλ j i ♣F i F j E j E i 1 si(λ) h1+λ j i ♣F j F i E i E j 1 si(λ) h1+λ j i F i F j E i E j 1 si(λ) h2+λ j i − 0 ♣ 1 si(λ) 0 (−1) λj c j,λ −(−1) λj c j,λ T ′ i j λ := E j E i F j F i 1 si(λ) h−λ j i ♣E i E j F j F i 1 si(λ) h1−λ j i ♣E j E i F i F j 1 si(λ) h1−λ j i E i E j F i F j 1 si(λ) h2−λ j i − 0 ♣ 1 si(λ) 0 (−t ij ) λi c −1 i,λ−αj (−1) λj c j,λ (−t ij ) λi c −1 i,λ−αj (−1) λj c j,λ T ′ i   j λ   := F j F i E j E i 1 si(λ) h−2−λ j i ♣F i F j E j E i 1 si(λ) h−1−λ j i ♣F j F i E i E j 1 si(λ) h−1−λ j i F i F j E i E j 1 si(λ) h−λ j i − 0 ♣ 1 si(λ) 0 (−t ij ) 1−λi c i,λ (−1) λj c −1 j,λ (−t ij ) 1−λi c i,λ (−1) λj c −1 j,λ 28 4. DEFINING THE SYMMETRY 2-FUNCTOR T ′ i,1 Again, the maps have the correct degree. For example, we have deg s i (λ) = 1+hj,s i (λ)i+1+hi,s i (λ)+α j i = 2+λ j −λ i (j·i)+λ i −λ i (i·i)+i·j = 1+λ j deg s i (λ) = 1+hi,s i (λ)i+1+hj,s i (λ)+α i i = 2+λ i −λ i (i·i)+λ j −λ i (j·i)+j·i = 1+λ j CHAPTER 5 Categorifying the Braid Group Action of T ′ i Chapter 4 explains how we define the categorified 2-functor T ′ i . In what follows, we prove that T ′ i does in fact satisfy the braid relations in the category Com(U). In this category, this amounts to showing that there exists homotopy equivalences between the complexesT ′ i T ′ j T ′ i (x) andT ′ j T ′ i T ′ j (x) and between T ′ i T ′ k (x) andT ′ k T ′ i (x) for all 1-morphismsx∈U Q when hi·ji =−1 andhi·ki = 0. On the decategorified level, the braiding of T i can be proved by using the relationship between T i and τ i , namely, T i (u) =τ i (u)τ −1 i for u∈ ˙ U and the fact that the τ i satisfy the braid relations. From this we see, for example, when hi·ji =−1 T i T j T i (u) =T i T j (τ i (u)τ −1 i ) =T i (τ j (τ i (u)τ −1 i )τ −1 j =τ i (τ j (τ i (u)τ −1 i )τ −1 j )τ −1 i T j T i T j (u) =T j T i (τ j (u)τ −1 j ) =T j (τ i (τ j (u)τ −1 j )τ −1 i =τ j (τ i (τ j (u)τ −1 j )τ −1 i )τ −1 j soT i T j T i (u) =T j T i T j (u). We use an analogousmethod inCom(U) to provethe braiding ofT i . At the categorified level the equationT i (u) =τ i (u)τ −1 i takes the formT i (u1 λ )τ i 1 λ ≃τ i u1 λ . This equivalence, whichwedenotebyi u1 λ ,isahomotopyequivalenceofchaincomplexeswhichweconstructexplicitlyfor all generating 1-morphismsu∈U. However, sinceτ i 1 λ = ∞ X b≥0 (−1) b F (λi+b) i E (b) i is an alternating sum of products of divided powers of Chevalley generators, the categorified versionofτ i 1 λ will need to involve a categorified version of divided powers which do not exist in the categoryU Q . We will therefore work in ˙ U Q , the Karoubi envelope of U Q . This enlarged category not only contains 1-morphisms E (b) i ,F (b) i which categorify the divided powers, but also graded 2-homs M t∈Z Hom(xhti,y) between 1-morphisms x,y ∈ ˙ U. The calculus governing the relations of these hom sets, the so-called “thick calculus” is best expressed as diagrams of “thick” strands. These thick strands are mixtures of the same thin strands that appear in the 2-homs of U Q so are ultimately controlled by the same relations. Still, when viewed as 2-morphisms of ˙ U Q a fascinating realm of combinatorialcomplexity presents itself. We briefly illustrate some of the basic ideas behind thick calculus, but the interested reader should consult [16] for a more in depth analysis of the interactions of these hom sets. 5.1. The Karoubi Envelope of U Q and Thick Calculus The Karoubi envelopeKar(M) of a categoryM is an enlargement ofM in which all idempotents split. An idempotent e: b→b in a categoryM is said to split if there exist morphisms b g b ′ h b such that e = hg and gh = Id b ′. More precisely, the Karoubi envelope Kar(M) is a category whose objects are pairs (b,e) where e: b→b is an idempotent of M and whose morphisms are triples of the form (e,f,e ′ ): (b,e)→ (b ′ ,e ′ ) 29 30 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i where f:b→b ′ inM making the diagram (5.1) b f e f b ′ e ′ b f b ′ commute. Thus, f must satisfy f = e ′ f = fe, which is equivalent to f = e ′ f = fe. Composition is induced from the composition inM, and the identity morphism is (e,e,e): (b,e)→ (b,e). WhenM is an additive category, the splitting of idempotents allows us to write (b,e)∈Kar(M) as im e, and we haveb ∼ = im e⊕im (Id b −e). The identity map Id b : b → b is an idempotent, and this gives rise to a fully faithful functor M → Kar(M). In Kar(M) all idempotents of M split and this functor is universal with respect to functors which split idempotents in M. When M is additive the inclusion M → Kar(M) is an additive functor (see [17, Section 9] and references therein). In the Karoubi envelope ˙ U Q we can define divided power 1-morphisms E (a) i 1 λ and F (b) i 1 λ by E (a) i 1 λ hti := E a i 1 λ D t− a(a−1) 2 E ,e a andF (a) i 1 λ hti := F a i 1 λ D t+ a(a−1) 2 E ,e ′ a wheretheidempotent e a is defined as e a :=δ a D a = • • a−2 a−1 • ... ... D a All strands are labelled i. Here D a is the longest braid on a-strands and the idempotents e ′ a are obtained frome a by a 180 ◦ rotation. The identity 2-morphism onE (a) i 1 λ is (5.2) (e a ,e a ,e a ) =: λ λ+2α i i a and identity 2-morphism onF (a) i is (e ′ a ,e ′ a ,e ′ a ) =: λ λ−2α i i a We define here some additional 2-morphisms in ˙ U Q which will appear often in the diagrammatics that follow. The degrees of these diagrams can be read from the shift on the right-hand side. a b λ a+b := e a e b b a λ :E (a+b) i 1 λ →E (a) i E (b) i 1 λ h−abi, a b λ a+b := e a+b a b λ :E (a) i E (b) i 1 λ →E (a+b) i 1 λ h−abi, 5.1. THE KAROUBI ENVELOPE OF UQ AND THICK CALCULUS 31 a b λ a+b := e ′ a+b a b λ :F (a+b) i 1 λ →F (a) i F (b) i 1 λ h−abi, a b λ a+b := e ′ a e ′ b b a λ :F (a) i F (b) i 1 λ →E (a+b) i 1 λ h−abi, n a := a e a e ′ a λ :E (a) i F (a) i 1 λ →1 λ ha 2 −aλ i i, n a := a e ′ a e a λ :F (a) i E (a) i 1 λ →1 λ ha 2 +aλ i i, n a := a e a e ′ a λ :1 λ →E (a) i F (a) i 1 λ ha 2 −aλ i i, n a := a e ′ a e a λ :1 λ →F (a) i E (a) i 1 λ ha 2 +aλ i i. In the proof that follows, many simplifications of diagrams will be necessary. Some of the most important relations used in these simplifications are listed below and will be referred to throughout the proof. For a detailed account of the derivation of these relations, see [16], [21]. 2-morphisms between 1-morphisms indexed by different nodes of the Dynkin diagram are repre- sented by strands of different colors. Only the relative positions of the nodes is important, so use the following convention for colors representing different combinations of node positions. Red strands will be labeled withi, black strandswithj, greenstrandswithk, and blue strandswithj ′ . In what follows, we will always assumehi,ji =hi,j ′ i =−1,hi,ki=hj,j ′ i = 0. 5.1.1. Thick Calculus Identities. Forhi,ji =−1 we have (5.3) a = a X k=0 t a−k ij t k ji a • •k ε a−k a = a X k=0 t a−k ij t k ji a • • k ε a−k (5.4) = + tij = + tij (5.5) = = 32 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i Square Flop (5.6) λ a b = − λ a b + X f1+f2+f3= λi+2b−1 λ • • • f1 f2 f3 a b There is a special case of the square flop that appears often enough that we show it here. λ a • j =          (−1) a c i,λ λ a j =λ i −a−1 0 j <λ i −a−1 Similarly, λ a • j =          c −1 i,λ λ a j =−λ i +a−1 0 j <−λ i +a−1 (5.7) • t 1 b − •t 1 b = X f+g=t−1 • f • g 1 b 5.2. Categorifying the Interaction between T ′ i and τ ′ i On the decategorified level of the quantum group, T ′ i (x1 λ )τ ′ i 1 λ = τ ′ i (x1 λ ). Categorifying this statement is a two step process, the first of which is showing that for any 1-morphismx inU Q we have a homotopy equivalence of complexes i x :T ′ i (x)τ ′ i 1 λ −→τ ′ i x1 λ (5.8) Where hereT ′ i is our categorified version of Lusztig’s T ′ i,1 andτ ′ i 1 λ is the complex (5.9) F (λi) i 1 λ h0i F (λi+1) i E (1) i 1 λ h1i F (λi+2) i E (2) i 1 λ h2i ... with differentiald b : (τ ′ i 1 λ ) b → (τ ′ i 1 λ ) b+1 given by d b := λi+b b . The second step is to show that these equivalences are natural with respect to the 2-functor T ′ i . In 5.3. CHAIN HOMOTOPY EQUIVALENCE iE k 33 other words, given a 2-morphism f : x → y we show that the following naturality square commutes (up to homotopy). (5.10) τ ′ i,1 y1 λ T ′ i (y)τ ′ i,1 1 λ τ ′ i,1 x1 λ T ′ i (x)τ ′ i,1 1 λ i y i x id τ ⊗f T ′ i (f)⊗id τ There are multiple cases for which these conditions must be checked so we briefly remind the reader of the order in which we complete this task. Equivalences i x must be computed for each of the Chevalley generators E i 1 λ ,F i 1 λ ,E j 1 λ ,F j 1 λ ,E k 1 λ ,F k 1 λ where hi,ji = −1 and hi,ki = 0. Once that is accomplished, we show 5.10 commutes for each generating 2-morphism f in U Q . This list includes the degree 2 dotted 2-morphism, degree -2 cross, and an adjoint cap cup pair for each of i,j, and k. In addition to these, we must check naturality on all possible mixed upward oriented crosses. We begin with the simplest cases, first constructingi E k ,i F k and then proving naturalityfor all generating 2-morphisms between k 1-morphisms. 5.3. Chain homotopy Equivalence i E k We compute i E k 1 λ : T ′ i (E k )τ ′ i 1 λ −→τ ′ i E k 1 λ hi,ki = 0 Beginning with E k , we have T ′ i,1 (E k 1 λ ) := 0 0 ♣E k 1 si(λ) therefore, T ′ i,1 (E k )τ ′ i,1 1 λ = E k F (λi) i 1 λ ... E k F (λi+b) i E (b) i 1 λ hbi ... d 0 d b−1 d b with d b = λi+b b . Since hi,ki = 0, E k F (λi+b) i E (b) i 1 λ hbi ∼ = F (λi+b) i E (b) i E k 1 λ hbi and we have an isomorphism (and thus homotopy equivalence) of chain complexes ... E k F (λi+b) i E (b) i 1 λ hbi E k F (λi+b+1) i E (b+1) i 1 λ hb+1i ... F (λi+b) i E (b) i E k 1 λ hbi F (λi+b+1) i E (b+1) i E k 1 λ hb+1i ... ... d b D b ∼ = ∼ = We choose the isomorphism r ik (λ) λi+b b , which has inverse r ik (λ) −1 t −b ik λi+b b . Here (and in what follows with the computations ofi Ej andi Fj ) we assign coefficientsr ik (λ) ands ik (λ) to the chain maps i E k and i F k . These coefficients, which depend onhi,ki andλ, will be pinned down in 34 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i the future so that the chain maps i E k and i F k satisfy all required naturality conditions. WecancalculateD b withsomediagramchasing. D b = t −b−1 ik λi+b b = λi+b b whichisprecisely the differential of the chain complex τ ′ i,1 E k 1 λ . Thus we have our desired homotopy equivalence of complexes i E k . T ′ i,1 (E k )τ ′ i,1 1 λ = ... E k F (λi+b) i E (b) i 1 λ hbi E k F (λi+b+1) i E (b+1) i 1 λ hb+1i τ ′ i,1 E k 1 λ = ... F (λi+b) i E (b) i E k 1 λ hbi F (λi+b+1) i E (b+1) i E k 1 λ hb+1i ... ... d b D b λi+b r ik (λ) b r ik (λ) b+1 5.4. Chain homotopy equivalence i F k . We compute i F k 1 λ T ′ i (F k )τ ′ i 1 λ −→τ ′ i F k 1 λ hi,ki = 0. T ′ i,1 (F k 1 λ ) := 0 0 ♣F k 1 si(λ) T ′ i,1 (F k )τ ′ i,1 1 λ = F k F (λi) i 1 λ ... F k F (λi+b) i E (b) i 1 λ hbi ... d 0 d b−1 d b with d b = λi+b b . Since F k F (λi+b) i E (b) i 1 λ hbi ∼ =F (λi+b) i E (b) i F k 1 λ hbi, we have an isomorphism (and thus homotopy equivalence) of chain complexes ... F k F (λi+b) i E (b) i 1 λ hbi F k F (λi+b+1) i E (b+1) i 1 λ hb+1i ... F (λi+b) i E (b) i F k 1 λ hbi F (λi+b+1) i E (b+1) i F k 1 λ hb+1i ... ... ∂ b D b ∼ = ∼ = Here, we choose the isomorphism s ik (λ)t −λ i −b ik λi+b b which has inverse s ik (λ) −1 λi+b b We then calculate D b to be t −λ i −b ik λi+b b = λi+b b which is precisely the differential of the 5.5. CHAIN HOMOTOPY EQUIVALENCE iE j 35 chain complex τ ′ i,1 F k 1 λ . Therefore, we have a homotopy equivalence of complexes T ′ i,1 (F k )τ ′ i,1 1 λ = ... F k F (λi+b) i E (b) i 1 λ hbi F k F (λi+b+1) i E (b+1) i 1 λ hb+1i τ ′ i,1 F k 1 λ = ... F (λi+b) i E (b) i F k 1 λ hbi F (λi+b+1) i E (b+1) i F k 1 λ hb+1i ... ... d b D b λi+b s ik (λ)t −λ i −b ik b s ik (λ)t −λ i −b−1 ik b+1 5.5. Chain homotopy equivalence i Ej T ′ i,1 (E j 1 λ ) = ♣E j E i 1 si(λ) E i E  1 si(λ) h1i j i and therefore T ′ i,1 (E j 1 λ )τ ′ i,1 1 λ = E j E i F (λi) i 1 λ ... [E i E j F (λi+b−1) i E (b−1) i ⊕E j E i F (λi+b) i E (b) i ]1 λ hbi ... d 0 d b−1 d b where the terms in the 0 th and b th homological degree are shown and the differentiald b is given by          − λi+b−1 b−1 λi+b b 0 λi+b b          For simplicity I will refer to the first summand in theb th homologicaldegree asM b and the second summand as N b . Using thick calculus relations in [16] it can be shown that M b ∼ = [F (λi+b−1) i E (b) i E j 1 λ M [b−1] F (λi+b−1) i E j E (b) i 1 λ M [b−1] F (λi+b−2) i E j E (b−1) i 1 λ ]hbi. (5.11) N b ∼ = [ M [b+1] F (λi+b) i E j E (b+1) i 1 λ M [b+1] F (λi+b−1) i E j E (b) i 1 λ ]hbi. (5.12) Again, for simplicity, we write M b ∼ =V b M A b M B b N b ∼ =C b M D b 36 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i The maps of the isomorphisms we will use are as follows: M b →V b M b →A b M b →B b λi+b−1 b−1 t 1−b ij (−1) λ i t 1−b ij (−1) λ i −k−1 • k λi+b−1 b−1 • β • γ • π ¯ x λi+b−1 b−1 (−1) λ i +b−1 X β,γ,¯ x V b →M b A b →M b B b →M b λi+b−1 b • π α λi+b−1 b • k λi+b−2 b−1 N b →C b N b →D b C b →N b D b →N b (−1) λ i +b+k • π 1 k λi+b b (−1) λ i +b X β,γ,¯ x • γ • β •π ¯ x λi+b b • k λi+b b+1 • k λi+b−1 b So we have an isomorphism of chain complexes (and homotopy equivalence) ... M b ⊕N b V b+1 ⊕A b+1 ⊕B b+1 ⊕C b+1 ⊕D b+1 ... V b ⊕A b ⊕B b ⊕C b ⊕D b M b+1 ⊕N b+1 ... ... d b ψ b+1 d b ψ −1 b ψ −1 b ψ −1 b+1 where ψ b is the isomorphism from M b ⊕N b to V b ⊕A b ⊕B b ⊕C b ⊕D b . The ordering of the summands above will effect the matrix of maps in the differential, so we choose a specific ordering of all summands so that the consequential steps involving Gaussian elimination are clear. The first two termsin the complexareslightlydifferent from allother terms, but the generalpattern is shownbelow. D 0 h0i⊕C 0 h0i−→V 1 ⊕D 1 h2i⊕C 1 h2i⊕C 1 h0i⊕D 1 h0i−→... (5.13) ...−→V b ⊕D b h+i⊕B b ⊕C b h+i⊕A b ⊕C b h0i⊕D b h0i−→... 5.5. CHAIN HOMOTOPY EQUIVALENCE iE j 37 Here, weuseC b h+ito representallC summandswith positivedegree. The firststep insimplifying this complexisto GaussianeliminateC 0 h0iandD 1 h0i. Oncethatisdone, weproceedbyeliminatingC 1 h0i andD 2 h0i. Wecontinueinthismanner,”inductively”eliminatingC b h0iandD b+1 h0i,whichispossible after the Gaussian elimination ofC b−1 h0i andD b h0i. The steps involved here are straightforward. We first show the the maps from C b h0i to D b+1 h0i are isomorphisms. Then we show that all maps from V b ⊕D b h+i⊕B b ⊕C b h+i⊕A b to D b+1 h0i are 0, i.e. that the matrix of maps in the differential from V b ⊕D b h+i⊕B b ⊕C b h+i⊕A b ⊕C b h0itoV b+1 ⊕D b+1 h+i⊕B b+1 ⊕C b+1 h+i⊕A b+1 ⊕C b+1 h0i⊕D b+1 h0i is upper block triangular. This will allow us to accurately track the remaining maps in the chain homotopic complexes to which we are simplifying by the homological algebra computation shown in (3.13). A bit of diagram chasing gives the following map between summands in C b and D b+1 . (5.14) (−1) λ i +b • k • γ • β •π ¯ x λi+b b+1 The degree of this map is 2(γ+β+|x|+k−b), and all morphisms fromC b h0i andD b+1 hti havek =b. So every morphism from C b h0i andD b+1 h0i will have the form (5.15) (−1) λ i +b c i,λ • b λi+b b+1 = (−1) λ i +b c i,λ • b λi+b b+1 = (−1) λ i +b c i,λ λi+b b+1 This is clearly an isomorphism. The simplifications of the diagram above can be made by using the thick calculus relations found in [16]. Next, WeexaminethemapsfromV b ⊕D b h+i⊕B b ⊕C b h+i⊕A b toD b+1 h0i. Itcanbeseenimmediately that there are no nonzero maps fromV b hbi⊕B b ⊕A b toD b+1 h0i so we must only check that all maps from C b h+i⊕D b h+i to D b+1 h0i are 0. The negative degree morphisms from C b h+i to D b+1 h0i have 38 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i the form • k λi+b b+1 with k<b. This simplifies to the 0 map. The negative degree maps from D b h+i to D b+1 h0i are • k = • k Which is also equal to 0 by proposition 5.2.9 in [16] since k<b. So this first phase of Gaussian elimination shows D 0 h0i⊕C 0 h0i−→V 1 ⊕D 1 h2i⊕C 1 h2i⊕C 1 h0i⊕D 1 h0i−→... ...−→V b ⊕D b h+i⊕B b ⊕C b h+i⊕A b ⊕C b h0i⊕D b h0i−→... is equivalent to the complex D 0 h0i−→V 1 ⊕D 1 h2i⊕C 1 h2i−→...−→V b ⊕D b h+i⊕B b ⊕C b h+i⊕A b −→... and the homotopy equivalence is just the injection of the summands of the latter complex into the summands of the former. Now we construct a complex which is equivalent to the complex above by a similar use of term by term Gaussian elimination. Beginning with the terms in 2nd and 3rd homological degree, we Gaussian eliminate C 1 h2i andA 2 . Once that is complete, we move to the terms in the 3rd and 4th homological degree and Gaussian eliminate C 2 h+i and A 3 . Again, we continue by ’inductively’ eliminating C b h+i andA b+1 until we have constructed the new homotopy equivalent complex D 0 h0i−→V 1 ⊕D 1 h2i−→...−→V b ⊕D b h+i⊕B b −→... In order for the homotopy equivalence to be the injection of the complex above into the complex D 0 h0i−→V 1 ⊕D 1 h2i⊕C 1 h2i−→...−→V b ⊕D b h+i⊕B b ⊕C b h+i⊕A b −→... we show that the differential fromV b ⊕D b h+i⊕B b ⊕C b h+i to V b+1 ⊕D b+1 h+i⊕B b+1 ⊕C b+1 h+i⊕A b+1 is a block upper triangular matrix with an invertible block of maps from C b h+i to A b+1 . The desired homotopy equivalence then results from (3.13). 5.5. CHAIN HOMOTOPY EQUIVALENCE iE j 39 The maps fromC b h+i to A b+1 are as follows with 0≤k ′ ≤b−1 and 0≤k≤b−1. (5.16) • k • k ′ λi+b b+1 These maps simplify to 0 if the degree is negative. The 0-degree maps simplify to the identity. Maps fromV b to A b+1 (5.17) • k λi+b+1 b These simplify to 0. Maps fromD b h+i to A b+1 . Here 0≤k ′ ≤b−1 and 0≤k≤b−1. (5.18) • k • k ′ λi+b−1 b 40 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i These simplify to 0. Maps fromB b to A b+1 . Here 0≤k ′ ≤b−1 and 0≤k≤b−2. (5.19) •k • k ′ λi+b−2 b−1 These simplify to 0. Finally, we repeat this process on the complex D 0 h0i−→V 1 ⊕D 1 h2i−→...−→V b ⊕D b h+i⊕B b −→... (5.20) byfirsteliminatingthesummandsD 1 h2iandB 2 . Fromthere,weeliminateD 2 h+iandB 3 andcontinue down the complex eliminating D b h+i and B b+1 each step of the way. In this way we construct the homotopy equivalent complex D 0 h0i−→V 1 h1i−→V 2 h2i−→...−→V b hbi−→... (5.21) This time, however, the differential in (5.20) is not upper block triangular, so the resulting homotopy equivalence will not just be an injection. The block of maps from D b h+i to B b+1 is invertible (shown below) so we may still use Gaussian Elimination. The morphism from D b hsi to B b+1 hti is (5.22) (−1) λ i +b X β+γ+¯ x= 1 2 t−1 k=b− 1 2 s • k • β • γ • π ¯ x λi+b−1 b = (−1) λi+b X β+γ+¯ x= 1 2 t−1 k=b− 1 2 s • k+β • • γ π ¯ x λi+b−1 b The degree of these maps is 2[(β+γ +|¯ x|+k)−(b−1)]. Furthermore if k+β <b−1 then the map simplifies to 0. For degree 0 morphisms we have β +γ +|¯ x| +k = b− 1 but all maps in this sum with γ +|¯ x| > 0 are zero since this forces k +β < b−1. Thus, we are left with the maps in which k+β =b−1 andγ,¯ x = 0, in which case, the map resolves to (−1) λi+b+λi+b−1 c i,λ+αj c −1 i,λ+αj Id =−Id. For negativedegree morphisms,k+β <b−1, so the negativedegree morphismsare all 0. We conclude that the matrix of morphisms fromD b h+i toB b+1 is upper triangular with−Id on the main diagonal and is therefore invertible. The diagram below illustrates the first step of Gaussian Elimination on the 2nd and 3rd terms of the 5.5. CHAIN HOMOTOPY EQUIVALENCE iE j 41 complex. D 0 h0i V 1 ⊕D 1 h+i V 2 ⊕D 2 h+i⊕B 2 D 0 h0i V 1 V 2 ⊕D 2 h+i h D 0 →V 1 D 0 →D 1 h+i i " V 1 →V 2 D 1 h+i→V 2 V 1 →D 2 h+i D 1 h+i→D 2 h+i V 1 →B 2 D 1 h+i→B 2 # [D 0 →V 1 ] h V 1 →V 2 V 1 →D 2 h+i i − h D 1 h+i→V 2 D 1 h+i→D 2 h+i i [D 1 h+i→B 2 ] −1 [V 1 →B 2 ] [1] 1 −[D 1 h+i→B 2 ] −1 [ V 1 →B 2 ] h 1 0 0 1 0 0 i Successively eliminating D b h+i with B b+1 leads to the following homotopy equivalent complexes (5.23) D 0 h0i V 1 ⊕D 1 h+i V 2 ⊕D 2 h+i⊕B 2 D 0 h0i V 1 V 2 ⊕D 2 h+i D 0 h0i V 1 V 2 [1] [1] 1 −[D 2 h+i→B 3 ] −1 [ V 2 →B 3 ] [D 0 →V 1 ] [V 1 →V 2 ]−[D 1 h+i→V 2 ][D 1 h+i→B 2 ] −1 [V 1 →B 2 ] h D 0 →V 1 D 0 →D 1 h+i i " V 1 →V 2 D 1 h+i→V 2 V 1 →D 2 h+i D 1 h+i→D 2 h+i V 1 →B 2 D 1 h+i→B 2 # [1] 1 −[D 1 h+i→B 2 ] −1 [ V 1 →B 2 ] h 1 0 0 1 0 0 i From this we first calculate the differential [V b →V b+1 ]−[D b h+i→V b+1 ][D b h+i→B b+1 ] −1 [V b →B b+1 ]. 42 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i Maps from D b h+i to V b+1 are (5.24) • k λi+b−1 b = • k λi+b−1 b This map reduces to 0 unless k = b, but all maps with source D b h+i have k ≤ b−1. Therefore, the matrix [D b h+i→V b+1 ] that appears in the differential is in fact the 0 matrix and the differential reduces to the map [V b →V b+1 ]. We calculate this morphism. (5.25) t −b ij (−1) λ i +1 = So we have constructed the complex (5.26) D 0 h0i V 1 V 2 V 3 ... which is homotopyequivalent toT ′ i,1 (E j )τ ′ i,1 1 λ . We note that the termsV b that appear in this complex are exactly the terms in the complex τ i,1 E j 1 λ and the map [V b →V b+1 ] is precisely the differential in the complex τ i,1 E j 1 λ . Therefore, the composition of the homotopy equivalences constructed in this section, will be a homotopy equivalence between τ ′ i,1 E j 1 λ and T ′ i,1 (E j )τ ′ i,1 1 λ . In order to evaluate the homotopy equivalence, we must determine the map −[D b h+i→B b+1 ] −1 [V b →B b+1 ]. Recall that the invertible matrix of maps from D b h+i to B b+1 was upper triangular with −Id morphisms along the diagonal. Therefore, the inverse matrix [D b h+i→B b+1 ] −1 will also be upper triangular with −Id 5.5. CHAIN HOMOTOPY EQUIVALENCE iE j 43 morphisms along the diagonal. Furthermore, maps fromV b hbi to B b+1 hti are (−1) λ i +b+1 X β+γ+¯ x= 1 2 t−1 • β • γ • π ¯ x λi+b−1 b = (−1) λ i +b+1 X β+γ+¯ x= 1 2 t−1 • β • • γ π ¯ x λi+b−1 b = X β+γ+¯ x= 1 2 t−1 • β• • γ π ¯ x λi+b−1 b The diagram above is 0 unlessβ≥b−1 However, the maximum value oft is 2b, soβ+γ+|¯ x|≤b−1 and it follows that the only nonzero morphism from V b hbi to B b+1 hti occurs when t = 2b, β = b−1 andγ =|¯ x| = 0. In this case the map simplifies to c −1 i,λ+α j λi+b−1 b So the matrix [V b →B b+1 ] is a column whose first component is the non-zero map above and all other components are 0. It follows that the product −[D b h+i→B b+1 ] −1 [V b →B b+1 ] will be the same column vector with coefficientc −1 i,λ+αj . From this point it is not too difficult to compose the homotopy equiva- lences in (5.23) with the isomorphismψ −1 to construct the following homotopy equivalent complexes. ... M b ⊕N b M b+1 ⊕N b+1 ... V b V b+1 ... ... d b D b i b Ej i b+1 Ej Here D b = λi+b−1 b and i b Ej =r ij (λ)                   λi+b−1 b c −1 i,λ+α j λi+b−1 b                   . Initially we note that i Ej will be an equivalence of complexes for any choice of invertible scalarr ij (λ), however, we will choose r ij (λ) later in order that certain relations with caps, cups, and crosses are preserved. 44 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i 5.6. Chain homotopy equivalence i Fj T ′ i (F j 1 λ ) = F j F i 1 si(λ) h−1i ♣F i F j 1 si(λ) j i so we haveT ′ i (F j 1 λ )τ ′ i,1 1 λ = F j F i F (λi) i 1 λ h−1i ... [F i F j F (λi+b) i E (b) i ⊕F j F i F (λi+b+1) i E (b+1) i ]1 λ hbi ... d −1 d b−1 d b where the terms in homological degree−1 andb are shown and the differential d b is given by          λi+b b λi+b+1 b+1 0 − λi+b+1 b+1          For simplicity we write M b :=F i F j F (λi+b) i E (b) i 1 λ hbi N b :=F j F i F (λi+b+1) i E (b+1) i 1 λ hbi By symmetrizing certain isomorphisms given in [16] we construct maps ϕ and ϕ −1 that give the following isomorphisms. M b ∼ =F (λi+b+1) i E (b) i F (1) j 1 λ hbi M [λi+b] F (1) j F (λi+b+1) i E (b) i 1 λ hbi := ¯ A b ⊕ ¯ B b N b ∼ = M [λi+b+2] F j F (λi+b+2) i E (b+1) i 1 λ hbi := ¯ C b The maps for ϕ b and ϕ −1 b are as follows: M b → ¯ A b M b → ¯ B b N b → ¯ C b λi+b b • β λi+b b •π α λi+b+1 b+1 ¯ A b →M b ¯ B b →M b ¯ C b →N b t −λ i −b ij λi+b+1 b t −λ i −b ij (−1) 1+|b α| • π b α λi+b+1 b (−1) α • α λi+b+2 b+1 5.6. CHAIN HOMOTOPY EQUIVALENCE iF j 45 whereα andβ depending on the degree of the source and target of the map. Using this isomorphism, have an isomorphism of chain complexes and in particular a homotopy equivalence. (5.27) ... M b ⊕N b ¯ A b+1 ⊕ ¯ B b+1 ⊕ ¯ C b+1 ... ¯ A b ⊕ ¯ B b ⊕ ¯ C b M b+1 ⊕N b+1 ... ... d b ϕ b+1 d b ϕ −1 b ϕ −1 b ϕ −1 b+1 We will calculate the maps in the differential matrix ϕ b+1 d b ϕ −1 b as needed. The ordering of the sum- mands above will effect the matrix of maps in the differential, and we choose a specific ordering of summands so that the consequential steps involving Gaussian elimination are clear. The first term in the complex is slightly different from all other terms, but the general pattern is shown below. e C −1 ⊕ ¯ C −1 h•i−→ e C 0 ⊕ ¯ A 0 ⊕ ¯ C 0 h•i⊕ ¯ B 0 −→... (5.28) ...−→ e C b ⊕ ¯ A b ⊕ ¯ C b h•i⊕ ¯ B b −→... Here we use e C b to represent the summand of ¯ C b with degree −λ i − 1 and ¯ C b h•i to represent the remaining ¯ C b summands in homological degree b. The degrees of these summands range over the values −λ i +1+2n for 0≤ n ≤ λ i +b. We show that the block of maps from ¯ C b h•i to ¯ B b+1 in the differential is actually an isomorphism. Then we use Gaussian elimination to construct the homotopy equivalent complex e C −1 −→ e C 0 ⊕ ¯ A 0 −→...−→ e C b ⊕ ¯ A b −→... (5.29) The differential in (5.28) is not upper triangular so we will have to take some care in computing the equivalence between (5.28) and (5.29). The maps from ¯ C b h•i to ¯ B b+1 have 0≤β,α≤λ i +b and degree = 2(β+α−λ i −b). (5.30) (−1) α • α • β λi+b+2 b+1 = (−1) α tji • • α+β λi+b+2 b+1 + (−1) α tij • α+β+1 λi+b+2 b+1 46 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i We note first that degree 0 maps have β +α = λ i +b in which case the diagram simplifies to the invertible map (5.31) tij(−1) λ i +b+1+α λi+b+2 b+1 Furthermore, when β +α<λ i +b, i.e. all negative degree maps, the diagram simplifies to 0. So the matrix of maps from ¯ C b h•i to ¯ B b+1 is upper triangular with invertible morphisms on the diagonal and is thus invertible. The first few terms of the first reduction of the complex using Gaussian elimination are shown below. For ease of computation of the homotopy equivalence, each block of maps in the differential has been replaced by the source and target of the corresponding morphisms. e C −1 ⊕ ¯ C −1 h•i e C 0 ⊕ ¯ A 0 ⊕ ¯ C 0 h•i⊕ ¯ B 0 e C 1 ⊕ ¯ A 1 ⊕ ¯ C 1 h•i⊕ ¯ B 1 e C −1 e C 0 ⊕ ¯ A 0 ⊕ ¯ C 0 h•i e C 1 ⊕ ¯ A 1 ⊕ ¯ C 1 h•i⊕ ¯ B 1 " e C −1 → e C 0 ¯ C −1 → e C 0 e C −1 → ¯ A 0 ¯ C −1 → ¯ A 0 e C −1 → ¯ C 0 ¯ C −1 → ¯ C 0 e C −1 → ¯ B 0 ¯ C −1 → ¯ B 0 # " e C 0 → e C 1 ¯ A 0 → e C 1 ¯ C 0 → e C 1 ¯ B 0 → e C 1 e C 0 → ¯ A 1 ¯ A 0 → ¯ A 1 ¯ C 0 → ¯ A 1 ¯ B 0 → ¯ A 1 e C 0 → ¯ C 1 ¯ A 0 → ¯ C 1 ¯ C 0 → ¯ C 1 ¯ B 0 → ¯ C 1 e C 0 → ¯ B 1 ¯ A 0 → ¯ B 1 ¯ C 0 → ¯ B 1 ¯ B 0 → ¯ B 1 # e C −1 → e C 0 e C −1 → ¯ A 0 e C −1 → ¯ C 0 − ¯ C −1 → e C 0 ¯ C −1 → ¯ A 0 ¯ C −1 → ¯ C 0 [ ¯ C −1 → ¯ B 0 ] −1 [ e C −1 → ¯ B 0 ] " e C 0 → e C 1 ¯ A 0 → e C 1 ¯ C 0 → e C 1 e C 0 → ¯ A 1 ¯ A 0 → ¯ A 1 ¯ C 0 → ¯ A 1 e C 0 → ¯ C 1 ¯ A 0 → ¯ C 1 ¯ C 0 → ¯ C 1 e C 0 → ¯ B 1 ¯ A 0 → ¯ B 1 ¯ C 0 → ¯ B 1 # 1 −[ ¯ C −1 → ¯ B 0 ] −1 [ e C −1 → ¯ B 0 ] 1 0 0 0 1 0 0 0 1 0 0 0 1 5.6. CHAIN HOMOTOPY EQUIVALENCE iF j 47 After repeatedly eliminating ¯ C b h•i with ¯ B b+1 we can construct the following homotopy equivalent complex. (5.32) e C −1 ⊕ ¯ C −1 h•i e C 0 ⊕ ¯ A 0 ⊕ ¯ C 0 h•i⊕ ¯ B 0 e C 1 ⊕ ¯ A 1 ⊕ ¯ C 1 h•i⊕ ¯ B 1 e C −1 e C 0 ⊕ ¯ A 0 e C 1 ⊕ ¯ A 1 " e C −1 → e C 0 ¯ C −1 → e C 0 e C −1 → ¯ A 0 ¯ C −1 → ¯ A 0 e C −1 → ¯ C 0 ¯ C −1 → ¯ C 0 e C −1 → ¯ B 0 ¯ C −1 → ¯ B 0 # " e C 0 → e C 1 ¯ A 0 → e C 1 ¯ C 0 → e C 1 ¯ B 0 → e C 1 e C 0 → ¯ A 1 ¯ A 0 → ¯ A 1 ¯ C 0 → ¯ A 1 ¯ B 0 → ¯ A 1 e C 0 → ¯ C 1 ¯ A 0 → ¯ C 1 ¯ C 0 → ¯ C 1 ¯ B 0 → ¯ C 1 e C 0 → ¯ B 1 ¯ A 0 → ¯ B 1 ¯ C 0 → ¯ B 1 ¯ B 0 → ¯ B 1 # h e C −1 → e C 0 e C −1 → ¯ A 0 i − h ¯ C −1 → e C 0 ¯ C −1 → ¯ A 0 i [ ¯ C −1 → ¯ B 0 ] −1 [ e C −1 → ¯ B 0 ] h e C 0 → e C 1 ¯ A 0 → e C 1 e C 0 → ¯ A 1 ¯ A 0 → ¯ A 1 i − h ¯ C 0 → e C 1 ¯ C 0 → ¯ A 1 i [ ¯ C 0 → ¯ B 1 ] −1 [ e C 0 → ¯ B 1 ¯ A 0 → ¯ B 1 ] 1 −[ ¯ C −1 → ¯ B 0 ] −1 [ e C −1 → ¯ B 0 ] 1 0 0 1 −[ ¯ C 0 → ¯ B 1 ] −1 [ e C 0 → ¯ B 1 ] −[ ¯ C 0 → ¯ B 1 ] −1 [ ¯ A 0 → ¯ B 1 ] 0 0 We first calculate the new differential since we will need to construct a homotopy equivalence between this new complex and the complex τ i F j 1 λ . We begin by examining the maps from ¯ C b to e C b+1 . • α λi+b+2 b+1 = • α λi+b+2 b+1 Since maps from ¯ C b have 0≤α≤λ i +b, all of these maps simplify to 0. Next we examine the maps from ¯ C b to ¯ A b . • α λi+b+2 b+1 = • α λi+b+2 b+1 Again, these simplify to 0 because 0≤α≤λ i +b. In other words, the matrix h ¯ C b → e C b+1 ¯ C b → ¯ A b+1 i that appears in the differential of (5.32) is the 0 matrix and the differential can be quickly calculated. 48 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i The map from e C b to e C b+1 is (5.33) (−1) λ i +b+2 • λi+b+1 λi+b+2 b+1 = (−1) λ i +b+2 • λi+b+1 λi+b+2 b+1 = −1 λi+b+2 b+1 The map from ¯ A b to e C b+1 is 0 because the differential from M b to N b+1 is 0. The map from e C b to ¯ A b+1 is (5.34) (−1) λ i +b+1 • λi+b+1 λi+b+2 b+1 = λi+b+2 b+1 The map from ¯ A b to ¯ A b+1 is (5.35) t λ i −b ij λi+b+1 b = λi+b+1 b So after all steps using Gaussian elimination are completed, we are left with the homotopy equiv- alent complex e C −1 e C 0 ⊕ ¯ A 0 e C 1 ⊕ ¯ A 1 e C 2 ⊕ ¯ A 2 ξ0 ξ−1 ξ1 5.6. CHAIN HOMOTOPY EQUIVALENCE iF j 49 Here ξ −1 =             − λi+1 λi+1             and ξ b =               − λi+b+2 b+1 0 λi+b+2 b+1 λi+b+1 b               for b≥ 0. This complex with this differential is homotopy equivalent to the complexτ i ⊗F j 1 λ as can be verified by the following commuting diagram of chain complexes. (5.36) e C −1 0 e C −1 0 e C 0 ⊕ ¯ A 0 e C 1 ⊕ ¯ A 1 e C 2 ⊕ ¯ A 2 ¯ A 2 e C 2 ⊕ ¯ A 2 ¯ A 2 ξ0 ¯ A 0 ¯ A 1 sij(λ) " 0 Id ¯ A 0 # sij(λ) " 0 Id ¯ A 1 # 0 0 0 sij(λ) " 0 Id ¯ A 2 # ω2 sij(λ) " 0 Id ¯ A 2 # e C 0 ⊕ ¯ A 0 e C 1 ⊕ ¯ A 1 ξ0 ¯ A 0 ¯ A 1 ω0 ω1 sij(λ) " 0 Id ¯ A 0 # sij(λ) " 0 Id ¯ A 1 # ξ−1 0 ξ−1 0 ξ1 ξ1 The map from e C b ⊕ ¯ A b to ¯ A b is ω b =    sij(λ) −1 c i,λ λi+b+2 b+1 sij(λ) −1 λi+b+1 b   .The homotopy from e C b ⊕ ¯ A b to e C b−1 ⊕ ¯ A b−1 is        c i,λ λi+b+2 b+1 0 0 0        and the homotopy from ¯ A b to ¯ A b−1 is 0. Sonowwecancomputethehomotopyequivalencefromτ i F j 1 λ toT ′ i (F j )τ i 1 λ bycomposingtheinjection of ¯ A into e C⊕ ¯ A from (5.36) with the homotopy equivalence from (5.32) and the isomorphismϕ −1 from (5.27). In terms of the source/target notation, this map will be the product of the matrices h 0 ¯ A b →M b 0 ¯ B b →M b e C b →N b 0 ¯ C b →N b 0 i 1 0 0 1 −[ ¯ C b → ¯ B b+1 ] −1 [ e C 0 → ¯ B 1 ] −[ ¯ C b → ¯ B b+1 ] −1 [ ¯ A b → ¯ B b+1 ] 0 0 0 sij(λ) = 50 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i s ij (λ) h 0 ¯ A b →M b 0 ¯ B b →M b e C b →N b 0 ¯ C b →N b 0 i 0 1 −[ ¯ C b → ¯ B b+1 ] −1 [ ¯ A b → ¯ B b+1 ] 0 = s ij (λ) h ¯ A b →M b −[ ¯ C b →N b ][ ¯ C b → ¯ B b+1 ] −1 [ ¯ A b → ¯ B b+1 ] i In order to calculate this map, we keep in mind the following facts: First of all, the maps from ¯ A b to ¯ B b+1 are 0 unless β = λ i +b in which case the nonzero map from ¯ A b hbi to ¯ B b+1 hλ i +2b+1i is t −λ i −b ij • β λi+b+1 b = t −λ i −b ij • β λi+b+1 b = t −λ i −b ij (−1) λ i +b λi+b+1 b Secondly, the isomorphism from ¯ C b → ¯ B b+1 was an upper triangular matrix with signed identity on the diagonal, so the inverse matrix will also be upper triangular with signed identity maps on the diagonal. Due to the previous calculation, we need be only concerned with the diagonal term from ¯ B b+1 hλ I +2b+1i to ¯ C b hλ i +2b+1i.Therefore, the matrix product [ ¯ C b → ¯ B b+1 ] −1 [ ¯ A b → ¯ B b+1 ] will be one nonzero map (shown above) in the first component followed by a column of zeros. From this we can easily compute the matrix product −[ ¯ C b → N b ][ ¯ C b → ¯ B b+1 ] −1 [ ¯ A b → ¯ B b+1 ] as it is just the composition of maps −[ ¯ C b hλ i +2b+1i → N b hbi][ ¯ B b+1 hλ i +2b+1i → ¯ C b hλ i +2b+1i][ ¯ A b hbi → ¯ B b+1 hλ i +2b+1i]. The diagram is shown below. t −λ i −b−1 ij λi+b+1 b 5.7. NATURALITY CONDITIONS 51 We now have the homotopy equivalence i b Fj :τ i F j 1 λ →T ′ i (F j )τ i 1 λ . The equivalence between the b th homological degrees of the complex is shown below. (5.37) sij(λ)                        t −λ i −b ij λi+b+1 b t −λ i −b−1 ij λi+b+1 b                        We make the choice for s ij (λ) later so that certain relations with caps and cups are preserved. 5.7. Naturality Conditions We now check that these homotopy equivalences satisfy the naturality condition with respect to the 2-functor T ′ i,1 . That is to say, we check 5.10 commutes for all generating 1-morphismsx involving E i , F i for alli∈I and corresponding generating 2-morphisms f. 5.7.1. k-naturality. Identity 2-morphism on E k . It’s easy to see that the diagram commutes when f is the identity 2-morphisms on E k . The 2-morphisms shown are the chain maps between the b th homological degrees of the respective complexes. τ ′ i,1 E k 1 λ T ′ i,1 (E k 1 λ )τ ′ i,1 1 λ τ ′ i,1 E k 1 λ T ′ i,1 (E k 1 λ )τ ′ i,1 1 λ r ik (λ) λi+b b r ik (λ) λi+b b b λi+b b λi+b The dot 2-morphism on E k . On the dot 2-morphisms we have τ ′ i,1 E k 1 λ T ′ i,1 (E k 1 λ )τ ′ i,1 1 λ τ ′ i,1 E k 1 λ T ′ i,1 (E k 1 λ )τ ′ i,1 1 λ r ik (λ) λi+b b r ik (λ) λi+b b b λi+b • b λi+b • Commutativity of this diagrams follows since dots slide through ani−k crossing whenhi,ki = 0. 52 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i Cross 2-morphismonE k E k . In order to construct the homotopyequivalence betweenτ ′ i,1 E k E k 1 λ andT ′ i,1 (E k E k 1 λ )τ ′ i,1 1 λ ,wecomposethehomotopyequivalencesfromτ ′ i,1 E k E k 1 λ toT ′ i,1 (E k 1 λ+α k )τ ′ i,1 E k 1 λ with that fromT ′ i,1 (E k 1 λ+α k )τ ′ i,1 E k 1 λ to T ′ i,1 (E k 1 λ+α k )T ′ i,1 (E k 1 λ )τ ′ i,1 1 λ . Using this equivalence, we can check that naturality holds for the 2-morphism :E k E k →E k E k . τ ′ i,1 E k E k 1 λ T ′ i,1 (E k E k 1 λ )τ ′ i,1 1 λ τ ′ i,1 E k E k 1 λ T ′ i,1 (E k E k 1 λ )τ ′ i,1 1 λ r ik (λ)r ik (λ+α k ) r ik (λ)r ik (λ+α k ) b λi+b b λi+b Repeated use of the R3 move shows that r ik (λ)r ik (λ+α k ) = r ik (λ)r ik (λ+α k ) so the square above commutes. Clockwise cap on E k F k . In order to check naturality on caps and cups, we first construct the homotopy equivalence between τ ′ i,1 E k F k 1 λ and T ′ i,1 (E k F k 1 λ )τ ′ i,1 1 λ . The naturality diagram for the clockwise cap is shown below. τ ′ i,1 1 λ T ′ i,1 (1 λ )τ ′ i,1 1 λ τ ′ i,1 E k F k 1 λ T ′ i,1 (E k F k 1 λ )τ ′ i,1 1 λ r ik (λ−α k )s ik (λ)t −λ i −b ik b λi+b t λ i ik b λi+b r ik (λ−α k )s ik (λ)t −b ik = r ik (λ−α k )s ik (λ) since we pick up a factor of t b ik from the i−k R2 move. Therefore, the squareabovewill commute aslong asthe conditionr ik (λ−α k )s ik (λ) = 1 is met. By a similar argument, one can show that naturality holds on the the counter-clockwisecup as long asr ik (λ)s ik (λ+α k ) = 1. This is the same condition as above with λ+α k in place of λ. 5.7. NATURALITY CONDITIONS 53 5.7.2. j naturality. We now check that the naturality square below commutes for generating 2-morphismsf :x→y involving 1-morphismsE j and F j . τ ′ i,1 y1 λ T ′ i,1 (y)τ ′ i,1 1 λ τ ′ i,1 x1 λ T ′ i,1 (x)τ ′ i,1 1 λ i y i x id τ ⊗f T ′ i,1 (f)⊗id τ Since we are working in Com(U), this boils down to checking that the difference of the chain maps that result from traversingthe commutative diagramabove is nullhomotopic. As we will see, in certain instances the chain maps will be equal to each other and 5.10 commutes on the nose, but with other caseswe will requirea chain homotopy. In both cases, wecheckthese conditions ontheb th homological degree of the chain complexes. identity 2-morphism on E j . We begin with the identity 2-morphism onE j . τ ′ i,1 E j 1 λ T ′ i,1 (E j 1 λ )τ ′ i,1 1 λ τ ′ i,1 E j 1 λ T ′ i,1 (E j 1 λ )τ ′ i,1 1 λ i b Ej i b Ej λi+b−1 b       λi+b−1 b−1 0 0 λi+b b       Clearly this diagram commutes. 54 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i The dot 2-morphism on E j the dot 2-morphism fromE j hti toE j ht+2i we have τ ′ i,1 E j 1 λ T ′ i,1 (E j 1 λ )τ ′ i,1 1 λ τ ′ i,1 E j 1 λ T ′ i,1 (E j 1 λ )τ ′ i,1 1 λ rij(λ)                   λi+b−1 b c −1 i,λ+α j λi+b−1 b                   rij(λ)                   λi+b−1 b c −1 i,λ+α j λi+b−1 b                   • λi+b−1 b       • λi+b−1 b−1 0 0 • λi+b b       which commutes since dots move through black-red crossings for free. Cross 2-morphism on E j E j . We must first construct the equivalence of complexes between τ i 1 λ+2αj E j E j 1 λ andT ′ i (E j E j 1 λ )τ i 1 λ beforewecancheckthatnaturalityholdsontheupwardj-cross. We accomplish this by composing equivalences, the one between τ i E j 1 λ+αj E j 1 λ and T ′ i (E j 1 λ )τ i 1 λ+αj E j 1 λ and the other between T ′ i (E j 1 λ )τ i 1 λ+αj E j 1 λ andT ′ i (E j E j 1 λ )τ i 1 λ . The former equivalence is 5.7. NATURALITY CONDITIONS 55 (5.38) r(i,j,λ+αj)                   λi+b−2 b c −1 i,λ+2α j λi+b−2 b                   and the latter is (5.39) rij(λ)                                          λi+b−2 b−1 0 c −1 i,λ+α j λi+b−2 b−1 0 0 λi+b−1 b 0 c −1 i,λ+α j λi+b−1 b                                          Composing gives the desired equivalence from τ i 1 λ+2αj ⊗E j E j 1 λ to T ′ i (E j E j 1 λ )⊗τ i 1 λ . We have i b EjEj = 56 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i (5.40) rij(λ+αj)rij(λ)                λi+b−2 b c −1 i,λ+α j λi+b−2 b c −1 i,λ+2α j λi+b−2 b c −1 i,λ+α j c −1 i,λ+2α j λi+b−2 b                T We now check that the following naturality square commutes. (5.41) τ ′ i E j E j 1 λ T ′ i (E j E j 1 λ )τ ′ i 1 λ τ ′ i E j E j 1 λ T ′ i (E j E j 1 λ )τ ′ i 1 λ i EjEj i EjEj T ′ i ( )⊗ The chainmapfromτ ′ i E j E j 1 λ toT ′ i (E j E j 1 λ )τ ′ i 1 λ hasfour components. We willcheckdiagramcom- mutativityonecomponentatatime,beginningwiththemapfromF (λi+b−2) i E (b) i E j E j 1 λ toE j E i E j E i F (λi+b) i E (b) i 1 λ . Moving counterclockwise around the commutative diagram 5.41 yields (5.42) c −1 i,λ+α j c −1 i,λ+2α j rij(λ)rij(λ+αj)t −1 ij λi+b−2 b = 5.7. NATURALITY CONDITIONS 57 (5.43) c −1 i,λ+α j c −1 i,λ+2α j rij(λ)rij(λ+αj) λi+b−2 b • + c −1 i,λ+α j c −1 i,λ+2α j rij(λ)rij(λ+αj)tjit −1 ij λi+b−2 b • The diagram without the dot on the i-strand is 0 and after some further simplification we are left with (5.44) c −1 i,λ+α j c −1 i,λ+2α j rij(λ)rij(λ+αj) λi+b−2 b which is precisely the map generated by moving clockwise around the diagram 5.41. Movingcounterclockwisearound5.41,themapfromF (λi+b−2) i E (b) i E j E j 1 λ toE i E j E j E i F (λi+b−1) i E (b−1) i 1 λ is (5.45) c −1 i,λ+α j rij(λ)rij(λ+αj) λi+b−2 b + c −1 i,λ+2α j rij(λ)rij(λ+αj)t −1 ij λi+b−2 b = 58 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i (5.46) c −1 i,λ+α j rij(λ)rij(λ+αj) λi+b−2 b The simplification results as we can move the upward j-cross in the first diagram through the thick i-lines and the second diagram simplifies to 0. The result is exactly the diagram we get by transversing the commutative diagram 5.41 clockwise. Movingcounterclockwisearound5.41,themapfromF (λi+b−2) i E (b) I E j E j 1 λ toE j E i E i E j F (λi+b−1) i E (b−1) i 1 λ is (5.47) c −1 i,λ+α j r(λ)r(λ+αj)t −1 ij λi+b−2 b + −c −1 i,λ+2α j rij(λ)rij(λ+αj)vij λi+b−2 b = (5.48) t −1 ij c −1 i,λ+α j rij(λ)rij(λ+αj) λi+b−2 b 5.7. NATURALITY CONDITIONS 59 This simplification results since the second diagram in the sum is 0. We can further simplify the remaining morphism with an i-j-i R3 move. This results is (5.49) t −1 ij c −1 i,λ+α j r(λ)r(λ+αj) λi+b−2 b − c −1 i,λ+α j rij(λ)rij(λ+αj) λi+b−2 b = (5.50) t −1 ij c −1 i,λ+α j r(λ)r(λ+αj) λi+b−2 b Traversing the commutative diagram clockwise yields the same map but with coefficient r ij (λ)r ij (λ+ α j )c −1 i,λ+2αj . However, due to a compatibility condition on our choice of scalars, we have c i,λ+αj c −1 i,λ = t ij , so c −1 i,λ+2αj =t −1 ij c −1 i,λ+αj and the diagrams match. The final component is the map fromF (λi+b−2) i E (b) I E j E j 1 λ toE i E j E i E j F (λi+b−1) i E (b−1) i . Traversing the commutative diagram 5.41 counter clockwise yields −r(λ)r(λ+αj)t −1 ij λi+b−2 b = −rij(λ)rij(λ+αj) • λi+b−2 b − t −1 ij tjirij(λ)rij(λ+αj) • λi+b−2 b 60 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i After R3 moves on i-strands and some thick calculus reductions, the diagram on the far right reduces to 0 and we are left with (5.51) rij(λ)rij(λ+αj) λi+b−2 b = r(λ)r(λ+αj) λi+b−2 b = rij(λ)rij(λ+αj) λi+b−2 b Here, the third equality results from a thick i-j R3 move ?? and is exactly the diagram that results from traversing 5.41 clockwise. Thus, on each of the four components of the b th degree of the chain maps we are comparing, we have equality and 5.41 commutes on the nose. clockwise j-cap. We must first construct the equivalence of complexes between τ i 1 λ E j F j 1 λ and T ′ i (E j F j 1 λ )τ i 1 λ before we can check that naturality holds on the clockwise cap. We accomplish this by composing equivalences, one between τ i E j 1 λ−αj F j 1 λ and T ′ i (E j 1 λ−αj )τ i 1 λ−αj F j 1 λ and the other betweenT ′ i (E j 1 λ−αj )τ i 1 λ−αj F j 1 λ and T ′ i (E j F j 1 λ )τ i 1 λ . The former equivalence is (5.52) rij(λ−αj)                   λi+b b c −1 i,λ λi+b b                   and the latter is 5.7. NATURALITY CONDITIONS 61 (5.53) sij(λ)                                       t −λ i −b+1 ij λi+b b−1 0 t −λ i −b ij λi+b b−1 0 0 t −λ i −b ij λi+b+1 b 0 t −λ i −b−1 ij λi+b+1 b                                       Composition of these equivalences yields (5.54) rij(λ−αj)sij(λ)               t −λ i −b+1 ij λi+b b t −λ i −b ij λi+b b c −1 i,λ t −λ i −b ij λi+b b c −1 i,λ t −λ i −b−1 ij λi+b b               T With this equivalence, we can now check the naturality condition on the clockwise cap, i.e. we show that the following diagram commutes. (5.55) τ ′ i,1 1 λ T ′ i,1 (1 λ )τ ′ i,1 1 λ τ ′ i,1 E j F j 1 λ T ′ i,1 (E j F j 1 λ )τ ′ i,1 1 λ i 1 λ i EjFj T ′ i,1 ( )⊗ 62 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i T ′ i,1 ( )⊗ =   0 (−1) λ j +1 c −1 j,λ (−1) λ j c −1 j,λ 0   So chasing the commutative diagram 5.55 counterclockwise gives the following maps on the b th homo- logical degree. (5.56) (−1) λ j +1 c −1 j,λ t −λ i −b ij rij(λ−αj)sij(λ) λi+b b + (−1) λ j c −1 j,λ c −1 i,λ t −λ i −b ij rij(λ−αj)sij(λ) λi+b b Thesecondmapequals0afterresolvingtheinneri-loop. ThefirstmapcanberesolvedusingamixedR2 move??andthethickcalculusrelation5.2.2from[16]. Lettinga = (−1) λj+1 c −1 j,λ t −λi−b ij r ij (λ−α j )s ij (λ) for the moment, we have (5.57) a λi+b b = a b−1 X k=0 t b−1−k ij t k ji • ε b−1−k • k λi+b b = a b−1 X k=0 (−1) λ i +b t b−1−k ij t k ji • ε b−1−k • k λi+b b The last diagram reduces to 0 except when b−1−k =b−1 or k = 0, in which case we have (5.58) a(−1) λ i −1 t b−1 ij λi+b b The map we get when traveling clockwise around 5.55 is the same except for the coefficient. So we force the conditiona(−1) λi−1 t b−1 ij = 1 so that naturality holds on the clockwise cap. This is equivalent to setting r ij (λ−α j )s ij (λ) = (−1) λi+λj c j,λ t λi+1 ij . 5.7. NATURALITY CONDITIONS 63 counter-clockwise j-cup. We construct the equivalence of complexes between τ i 1 λ F j E j 1 λ and T ′ i (F j E j 1 λ )τ i 1 λ in the same fashion as we did the equivalence of the previous section, i.e. by com- posing equivalences, one betweenτ i F j 1 λ+αj E j 1 λ andT ′ i (F j 1 λ+αj )τ i 1 λ+αj E j 1 λ , and the other between T ′ i (F j 1 λ+αj )τ i 1 λ−αj E j 1 λ andT ′ i (F j E j 1 λ )τ i 1 λ . The former equivalence is (5.59) s ij (λ+α j )                t −λ i −b+1 ij λi+b b t −λ i −b ij λi+b b                and the latter is (5.60) r ij (λ)                                           λi+b−1 b 0 λi+b−1 b c −1 i,λ+α j 0 0 λi+b b+1 0 λi+b b+1 c −1 i,λ+α j                                           Composing these gives the equivalence i FjEj inb th homological degree. i b FjEj = 64 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i (5.61) e c               λi+b b t −λ i −b+1 ij λi+b b t −λ i −b+1 ij c −1 i,λ+α j λi+b b t −λ i −b ij λi+b b t −λ i −b ij c −1 i,λ+α j               T where the initial constante c =r ij (λ)s ij (λ+α j ). With this equivalence in hand, we check that the following diagram commutes. (5.62) τ ′ i,1 F j E j 1 λ T ′ i,1 (F j E j 1 λ )τ ′ i,1 1 λ τ ′ i,1 1 λ T ′ i,1 (1 λ )τ ′ i,1 1 λ i FjEj i 1 λ T ′ i,1 ( )⊗ Moving counterclockwisearound 5.62 we have the following maps betweenb th homological degrees T ′ i,1 ( )⊗ =   0 (−1) λ j c j,λ (−1) λ j +1 c j,λ 0   Moving clockwise around 5.62 produces (5.63) e c               λi+b b t −λ i −b+1 ij λi+b b t −λ i −b+1 ij c −1 i,λ+α j λi+b b t −λ i −b ij λi+b b t −λ i −b ij c −1 i,λ+α j               T so clearly, 5.62 does not commute on the nose. We will require that these chain maps be chain homotopic. Next we construct the chain null-homotopy, H, such that the difference of these chain 5.7. NATURALITY CONDITIONS 65 maps is null-homotopic. Diagrammatically we have (5.64) T ′ i,1 (F j E j 1 λ )τ ′ i,1 1 b−1 λ T ′ i,1 (F j E j 1 λ )τ ′ i,1 1 b λ T ′ i,1 (F j E j 1 λ )τ ′ i,1 1 b+1 λ τ ′ i,1 1 b−1 λ τ ′ i,1 1 b λ τ ′ i,1 1 b+1 λ d b−1 d b λi+b b D b = f b −g b H b H b+1 with the homotopy conditiond b−1 H b +H b+1 D b =f b −g b for all b, where f andg represent the chain maps from 5.62,D b is the differential of the complex τ ′ i,1 1 λ and the differentiald b = (5.65)                       − 0 0 0 0 0 − 0 0 0 −                       There are 4 summands in each homologicaldegree ofT ′ i,1 (F j E j 1 λ )τ ′ i,1 1 λ , so the homotopyH b will have 4 components. We proveH b = [h b 2 h b 3 h b 4 h b 1 ] T where 66 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i h b 1 = 0 h b 2 = (−1) λi+λj+1 c j,λ b−1 X k=0 (−1) k+1 • • k ε b−1−k λi+b b h b 3 = (−1) λi+λj+1 c j,λ b−2 X k=0 b−2−k X l=0 t k ji t −k−1 ij (−1) l+1 • • • l ε b−2−k−l k λi+b b h b 4 = (−1) λi+λj+1 c j,λ b−1 X k=0 b−1−k X l=0 t k ji t −k−1 ij (−1) l+1 • • • l ε b−1−k−l k λi+b b + • • • l ε b−1−k−l k λi+b b One can prove this is a homotopy simply by checking that the homotopy condition in 5.64 is satisfied on each component, however, the derivation of this homotopy is instructive, so we illustrate its derivation with h 4 . First we track the homotopy condition on h b 4 in 5.64. This yields (5.66) h b 4 − + h b+1 4 = e ct −λ i −b ij c −1 i,λ+α j λi+b b The right hand side of this equation can be resolved into the following sum. (5.67) e c b X k=0 b−k X l=0 (−1) l t k ji t λi−k ij c −1 i,λ+αj        • • • l ε b−k−l k λi+b b + l−1 X p=0 • • • • l−1−p ε b−k−l k p λi+b b        From here, we identify the maps in this sum from which we can ’factor’ out the thick differential from the bottom and we hypothesize that the remaining portion of this map is h b+1 4 . In this case, this 5.7. NATURALITY CONDITIONS 67 leads to the assumption that (5.68) h b+1 4 =e c b X k=0 b−k X l=0 (−1) l t k ji t λi−k ij c −1 i,λ+αj • • • l ε b−k−l k λi+b+1b+1 . We then must show that the remaining maps in the sum 5.67 will result from the composition (5.69) h b 4 − , so that equation 5.66 is satisfied. In this case, this amounts to showing e c b−1 X k=0 b−1−k X l=0 (−1) l+1 t k ji t λi−k ij c −1 i,λ+αj • • • l ε b−k−l−1 k λi+b b = e c b X k=0 b−k X l=0 l−1 X p=0 (−1) l t k ji t λi−k ij c −1 i,λ+αj • • • • l−1−p ε b−k−l k p λi+b b . (5.70) This can be shown to be true using a few thick calculus relations and the combinatorial identity (5.71) m X l=0 m−l X p=0 f(l,p)= m X r=0 r X p=0 f(r−p,p). This proves that the formula for h b 4 is correct. (The presentation of h b 4 given earlier can be simplified to the form above by using the square flop relation.) Now that we know h b 4 , this entire process can be repeated with h b 2 and h b 3 . For example, the homotopy condition for h b 3 is (5.72) h b 3 + h b+1 3 − h b 4 = e ct −λ i −b ij λi+b b + (−1) λ j c j,λ . 68 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i We plug in for h b 4 and combine and resolve the known maps on the right hand side of the equation. Grouping all terms of the resulting sum from which the thick differential can be factored from the bottom allows us to make a guess at what the maph b+1 3 will be. From there we plug h b 3 into 5.72 and show that equality holds in 5.72. This process is then used to calculate h b+1 2 and h b+1 1 in that order. During the computation of h b 3 and h b 2 , however, it becomes apparent that the homotopy H will only work ife c =:r ij (λ)s ij (λ+α j ) = (−1) λi+λj+1 t λi ij c j,λ . This is equivalent to the condition that arose in the previoussection, so weforcethis conditionto holdand conclude the naturalitycondition5.62 holds with the counter clockwise cup. 5.7.3. Mixed-node naturality. E j ′E j cross We now check the naturality condition on theE j ′E j cross, where hi,ji = hi,j ′ i = −1. We use blue strands to represent j’ strands and black for j strands. In order to construct the equivalence between τ i 1 λ+α j ′+αj E j ′E j 1 λ and T ′ i (E j ′E j 1 λ )τ i 1 λ , we compose equivalences, one between τ i E j ′1 λ+αj E j 1 λ and T ′ i (E j ′1 λ )τ i 1 λ+αj E j 1 λ and the other between T ′ i (E j ′1 λ )τ i 1 λ+αj E j 1 λ and T ′ i (E j ′E j 1 λ )τ i 1 λ . The former equivalence is (5.73) r ij ′(λ+αj)                   λi+b−2 b c −1 i,λ+α j +α j ′ λi+b−2 b                   5.7. NATURALITY CONDITIONS 69 and the latter is (5.74) rij(λ)                                          λi+b−2 b−1 0 c −1 i,λ+α j λi+b−2 b−1 0 0 λi+b−1 b 0 c −1 i,λ+α j λi+b−1 b                                          Composing gives the desired equivalence fromτ i 1 λ+α j ′+αj E j ′E j 1 λ to T ′ i (E j ′E j 1 λ )τ i 1 λ . i b E j ′Ej = (5.75) r ij ′(λ+α j )r ij (λ)                λi+b−2 b c −1 i,λ+α j λi+b−2 b c −1 i,λ+α j +α j ′ λi+b−2 b c −1 i,λ+α j c −1 i,λ+α j +α j ′ λi+b−2 b                T We now check that the following naturality square commutes. (5.76) τ ′ i,1 E j E j ′1 λ T ′ i,1 (E j E j ′1 λ )τ ′ i,1 1 λ τ ′ i,1 E j ′E j 1 λ T ′ i,1 (E j ′E j 1 λ )τ ′ i,1 1 λ i EjE j ′ i E j ′Ej T ′ i,1 ( )⊗ 70 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i The chain map from τ ′ i,1 E j ′E j 1 λ to T ′ i,1 (E j E j ′1 λ )τ ′ i,1 1 λ has four components. We will check dia- gram commutativity one component at a time, beginning with the map from F (λi+b−2) i E (b) i E j ′E j 1 λ toE j E i E j ′E i F (λi+b) i E (b) i 1 λ . Moving counterclockwise around the commutative diagram 5.76 yields (5.77) c −1 i,λ+α j c −1 i,λ+α j +α j ′ r(i,j ′ ,λ+αj)r(i,j,λ)t −1 ij ′ λi+b−2 b = (5.78) c −1 i,λ+α j c −1 i,λ+α j +α j ′ r ij ′(λ+αj)rij(λ)t −1 ij ′ tij λi+b−2 b • + c −1 i,λ+α j c −1 i,λ+α j +α j ′ r ij ′(λ+αj)rij(λ)t −1 ij ′ tji λi+b−2 b • The diagram without the dot on the i-strand is 0 and after using the relationship between compatible scalars, we are left with (5.79) c −1 i,λ+α j ′ c −1 i,λ+α j +α j ′ r(i,j ′ ,λ+αj)r(i,j,λ) λi+b−2 b 5.7. NATURALITY CONDITIONS 71 Movingcounterclockwisearound5.76,themapfromF (λi+b−2) i E (b) i E j ′E j 1 λ toE i E j E j ′E i F (λi+b−1) i E (b−1) i 1 λ is (5.80) t −1 ij ′ tijc −1 i,λ+α j r ij ′(λ+αj)rij(λ) λi+b−2 b + t −1 ij ′ c −1 i,λ+α j +α j ′ r ij ′(λ+αj)rij(λ) λi+b−2 b = (5.81) t −1 ij ′ tijc −1 i,λ+α j r ij ′(λ+αj)rij(λ) λi+b−2 b Movingcounterclockwisearound5.76,themapfromF (λi+b−2) i E (b) I E j ′E j 1 λ toE j E i E i E j ′F (λi+b−1) i E (b−1) i 1 λ is (5.82) t −1 ij ′ c −1 i,λ+α j r ij ′(λ+αj)rij(λ) λi+b−2 b = t −1 ij ′ c −1 i,λ+α j r ij ′(λ+αj)rij(λ) λi+b−2 b 72 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i We can further simplify this morphism with an i-j-i R3 move. This results is (5.83) c −1 i,λ+α j r ij ′(λ+αj)rij(λ)t −1 ij ′ λi+b−2 b − c −1 i,λ+α j r ij ′(λ+αj)rij(λ) λi+b−2 b = (5.84) t −1 ij ′ c −1 i,λ+α j r ij ′(λ+αj)rij(λ) λi+b−2 b ThefinalcomponentisthemapfromF (λi+b−2) i E (b) i E j ′E j 1 λ toE i E j E i E j ′F (λi+b−1) i E (b−1) i . Traversing the commutative diagram 5.76 counter clockwise yields (5.85) −t −1 ij ′ r ij ′(λ+αj)rij(λ) λi+b−2 b = 5.7. NATURALITY CONDITIONS 73 (5.86) −r ij ′(λ+αj)rij(λ) • λi+b−2 b − t −1 ij ′ t j ′ i r ij ′(λ+αj)rij(λ) • λi+b−2 b After R3 moves on i-strands and some thick calculus reductions, the diagram on the far right reduces to 0 and we are left with (5.87) r ij ′(λ+αj)rij(λ) λi+b−2 b = r ij ′(λ+αj)rij(λ) λi+b−2 b = r ij ′(λ+αj)rij(λ) λi+b−2 b When we traverse the commutative diagram 5.76 clockwise, we obtain the exact same morphisms on each of the four components, but there are different coefficients on eachcomponent. We note, however, that if we impose the conditionr ij ′(λ+α j )r ij (λ) =r ij (λ+α j ′)r ij ′(λ) then 5.76 commutes on the nose. E k E j cross. We now check the naturality condition on the E k E j cross, where hi,ji = −1 and hi,ki = 0. We use green strands to represent k strands and black for j strands. (5.88) τ ′ i,1 E j E k 1 λ T ′ i,1 (E j E j 1 λ )τ ′ i,1 1 λ τ ′ i,1 E k E j 1 λ T ′ i,1 (E k E j 1 λ )τ ′ i,1 1 λ i EjE k i E k Ej T ′ i ( )⊗ 74 5. CATEGORIFYING THE BRAID GROUP ACTION OF T ′ i We will need bothi E k Ej andi EjE k . The computationsare straightforwardso we givethese chain maps below. (5.89) i b EjE k = r ik (λ)rij(λ+α k )                   λi+b−1 b c −1 i,λ+α k +α j λi+b−1 b                   (5.90) i b E k Ej = rij(λ)r ik (λ+αj)                   λi+b−1 b c −1 i,λ+α j λi+b−1 b                   Equating the maps that result from traversing 5.88 in both directions gives rij(λ)r ik (λ+αj)                               λi+b−1 b c −1 i,λ+α j λi+b−1 b                               = r ik (λ)rij(λ+α k )                               λi+b−1 b c −1 i,λ+α k +α j λi+b−1 b                               This leads to the equations of coefficients r ij (λ)r ik (λ+α j )t ik = r ij (λ+α k )r ik (λ) and r ij (λ)r ik (λ + α j )c −1 i,λ+αj = r ij (λ+α k )r ik (λ)c −1 i,λ+α k +αj . These equations are equivalent due to the compatibility of scalars, so we force the relationshipr ij (λ)r ik (λ+α j )t ik =r ij (λ+α k )r ik (λ) to hold. 5.7. NATURALITY CONDITIONS 75 E j E k cross. Checking naturality for the E j E k cross is similarly accomplished and leads to the following equality of maps. r ik (λ)rij(λ+αj)                               λi+b−1 b c −1 i,λ+α k +α j λi+b−1 b                               = rij(λ)r ik (λ+αj)                               λi+b−1 b c −1 i,λ+α j λi+b−1 b                               Equating these diagrams also leads to the equations r ij (λ)r ik (λ +α j )t ik = r ij (λ +α k )r ik (λ) and r ij (λ)r ik (λ+α j )c −1 i,λ+αj =r ij (λ+α k )r ik (λ)c −1 i,λ+α k +αj determined in the previoussection. So assuming a suitable choice of coefficients satisfies r ij (λ)r ik (λ+α j )t ik = r ij (λ+α k )r ik (λ), naturality holds for the E j E k cross. 5.7.4. Determining chain map coefficients. In previous sections we determined relationships between the chain map coefficients. These are listed below. k-caps and k-cups r ik (λ−α k )s ik (λ) = 1 j-caps and j-cups r ij (λ−α j )s ij (λ) = (−1) λi+λj t λi+1 ij c j,λ E j E j ′ cross r ij (λ+α ′ j ) r ij (λ) = r ij ′(λ+α j ) r ij ′(λ) E j E k cross r ij (λ+α k ) r ij (λ) =t ik r ik (λ+α j ) r ik (λ) There are many options that one could choose for these coefficients, but perhaps the simplest choice is r ij (λ) = 1 hi,ji =−1 s ij (λ) = (−1) λi+λj t λi+1 ij c j,λ hi,ji =−1 r ik (λ) =t λi ik hi,ki = 0 s ik (λ) =t −λi ik hi,ki = 0 CHAPTER 6 The Categorified Braid Group Action 6.1. The Braiding of T ′ i We first give a quick summary of what we have proved in this paper thus far and then follow with a diagrammatic proof thatT ′ i satisfies the braid relations.There are two important facts aboutτ ′ i that we use in the proof. We know τ ′ i is invertible. In other words, there exists τ ′−1 i ∈Com( ˙ U Q ) such that the tensor product of complexesτ ′ i τ ′−1 i is homotopy equivalent to the identity complex. We also know that for each i ∈ I, the family of complexes τ ′ i satisfy the braid relations. For example, there is a homotopy equivalence between the complexesτ ′ i τ ′ j τ ′ i andτ ′ j τ ′ i τ ′ j whenhi,ji =−1. We have defined the 2-endofunctorT ′ i onCom( ˙ U Q ). Furthermore, we proved that for every 1-morphismx in ˙ U Q , we have a natural isomorphism i x : τ ′ i x ∼ − →T ′ i (x)τ ′ i . Using the fact that τ ′ i is invertible, this leads to the natural isomorphism Ψ −1 i (x): τ ′ i xτ −1 i → T ′ i (x) and its inverse Ψ i (x): T ′ i (x) → τ ′ i xτ −1 i for any 1-morphism x in ˙ U Q . The following diagram is a useful device that helps illustrate the interactions of these maps on the level of objects, 1-morphisms, and 2-morphisms in ˙ U Q . s i (ν) ν λ s i (λ) τi −1 x τi Ti(x) Ψi(x) Tensoring the naturality diagram 5.10 with τ ′−1 i leads to the commutative diagram (6.1) τ ′ i yτ ′−1 i 1 si(λ) T ′ i (y)1 si(λ) τ ′ i xτ ′−1 i 1 si(λ) T ′ i (x)1 si(λ) Ψ −1 i (y) Ψ −1 i (x) id τ ⊗f⊗id τ −1 T ′ i (f) 76 6.1. THE BRAIDING OF T ′ i 77 which can be diagrammatically depicted by equating the diagrams s i (ν) ν λ s i (λ) τi −1 y τi Ti(y) Ti(x) Ψi(y) Ti(f) = s i (ν) ν λ s i (λ) τi −1 y x τi Ti(x) f Ψi(x) (6.2) When hi,ji =−1,Composing the diagrams for Ψ and Ψ −1 in a specific order enables us to define the isomorphismβ jij iji : T i T j T i (x)→T j T i T j (x). Diagrammatically we have sisjsi(ν) sisj(ν) sj(ν) ν λ sj(λ) sisj(λ) sisjsi(λ) sjsi(λ) si(λ) si(ν) sjsi(ν) x Tj(x) TiTj(x) TjTiTj(x) Ti(x) TjTi(x) TiTjTi(x) τ −1 j τ −1 i τ −1 j τ −1 i τ −1 j τ −1 i τj τi τj τi τj τi Ψi(x) Ψj(Ti(x)) Ψi(TjTi(x)) Ψ −1 j (x) Ψ −1 i (Tj(x)) Ψ −1 j (TiTj(x)) Note how the “eyes” of the diagram depict the braid relationτ i τ j τ i ≃τ j τ i τ j . The isomorphismβ jij iji is also natural in the sense that for a 2-morphismf: x→y, the diagram below commutes. T i T j T i (x) T i T j T i (y) T j T i T j (y) T j T i T j (x) TjTiTj(f) β jij iji (y) β jij iji (x) TiTjTi(f) (6.3) 78 6. THE CATEGORIFIED BRAID GROUP ACTION The composition resulting from moving clockwise around this commuting square can be depicted by sisjsi(ν) sisj(ν) sj(ν) ν λ sj(λ) sisj(λ) sisjsi(λ) sjsi(λ) si(λ) si(ν) sjsi(ν) y Tj(y) TiTj(y) TjTiTj(y) Ti(y) TjTi(y) TiTjTi(y) τ −1 j τ −1 i τ −1 j τ −1 i τ −1 j τ −1 i τj τi τj τi τj τi Ψi(y) Ψj(Ti(y)) Ψi(TjTi(y)) Ψ −1 j (y) Ψ −1 i (Tj(y)) Ψ −1 j (TiTj(y)) TiTjTi(x) TiTjTi(f) 6.2. NEW PROJECTS 79 Using multiple “substitutions” of the diagram equality 6.2 into this diagram allows us to move the 2-morphismf up through the diagram until we reach the diagram sisjsi(ν) sisj(ν) sj(ν) ν λ sj(λ) sisj(λ) sisjsi(λ) sjsi(λ) si(λ) si(ν) sjsi(ν) x Tj(x) TiTj(x) TjTiTj(x) Ti(x) TjTi(x) TiTjTi(x) τ −1 j τ −1 i τ −1 j τ −1 i τ −1 j τ −1 i τj τi τj τi τj τi Ψi(x) Ψj(Ti(x)) Ψi(TjTi(x)) Ψ −1 j (x) Ψ −1 i (Tj(x)) Ψ −1 j (TiTj(x)) TjTiTj(y) TjTiTj(α) This diagram depicts the counterclockwise path around 6.3 thus proving that β jij iji respects the natu- rality condition. The proof that there exists isomorphism β ki ik : T i T k → T k T i when hi,ki = 0 is proved in a similar fashion. This completes the proof that the categorified Lusztig operatorT i satisfies the braid relations up to 2-natural isomorphism. We note here that once we define the categorified Lusztig operators T ′′ i,1 ,T ′ i,−1 , and T ′′ i,−1 using Abram’s categorified symmetries, the proof that each family of operators satisfy the braid relations is completed in an almost identical fashion. 6.2. new projects There is a well known construction that leads to the PBW basis for a universal enveloping algebra U(g). The construction utilizes the Weyl Group action on a set of simple root vectors in order to generate a set of linearly independent vectors that span the positive root spaces. There is a similar construction that generates a PBW type basis for the quantum group U q (g), but the action of the Weyl group is replaced by the action of the Internal braid group action of the T i,e . In categorifying this braid group action, we have provided a tool that enables us to raise this PBW basis construction to the level of the categorified quantum group. Using an analogous construction to the decategorified case, we can construct certain modules of the KLR algebra that lift PBW basis vectorsof the quantum group. In fact, this project is already underway. McNamara is constructing projective resolutions of standard modules using the categorified braid group action defined in this thesis. This will hopefully augment the already existing rich theory of standard modules. Bibliography [1] A.Beilinson, G. Lusztig, and R. MacPherson. Ageometric setting forthe quantum deformation of GLn. Duke Math. J., 61(2):655–677, 1990. [2] A. Beliakova, K. Habiro, A. Lauda, and B. Webster. Cyclicity for categorified quantum groups. 2015. arXiv: UP- DATE ARXIV NUMBER. [3] A. Beliakova, M. Khovanov, and A. D. Lauda. A categorification of the Casimir of quantum sl(2). Adv. Math., 230(3):1442–1501, 2012. arXiv:1008.0370,. [4] Anna Beliakova, Kazuo Habiro, Aaron D. Lauda, and Ben Webster. Cyclicity for categorified quantum groups. J. Algebra, 452:118–132, 2016. [5] J. Brundan. Symmetric functions, parabolic categoryO, and the Springer fiber. Duke Math. J., 143(1):41–79, 2008. arXiv:math/0608235. [6] J. Brundan. On the definition of Kac-Moody 2-category, 2015. arxiv:1501.00350. [7] J. Brundan and C. Stroppel. Highest weight categories arising from Khovanov’s diagram algebra III: category O. Rep. Theory, 15:170–243, 2011. arXiv:0812.1090. [8] S. Cautis. Rigidity in higher representation theory. 2014. arXiv:1409.0827. [9] S. Cautis and A. D. Lauda. Implicit structure in 2-representations of quantum groups. Selecta Mathematica, pages 1–44, 2014. arXiv:1111.1431. [10] Sabin Cautisand Joel Kamnitzer. Braidingviageometric Liealgebra actions. Compos. Math., 148(2):464–506, 2012. [11] Sabin Cautis, Joel Kamnitzer, and Scott Morrison. Webs and quantum skew Howe duality. Math. Ann., 360(1- 2):351–390, 2014. [12] J. Chuang and R. Rouquier. Derived equivalences for symmetric groups and sl 2 -categorification. Ann. of Math. (2), 167(1):245–298, 2008. [13] M. Khovanov and A. Lauda. A diagrammatic approach to categorification of quantum groups I. Represent. Theory, 13:309–347, 2009. arXiv:0803.4121. [14] M.KhovanovandA.Lauda.Adiagrammaticapproachtocategorification ofquantumgroupsIII.Quantum Topology, 1:1–92, 2010. arXiv:0807.3250. [15] M. Khovanov and A. Lauda. A diagrammatic approach to categorification of quantum groups II. Trans. Amer. Math. Soc., 363:2685–2700, 2011. arXiv:0804.2080. [16] M. Khovanov, A. Lauda, M. Mackaay, and M. Stoˇ si´ c. Extended graphical calculus for categorified quantum sl(2). Memoirs of the AMS, 219, 2012. arXiv:1006.2866. [17] A. D. Lauda. A categorification of quantum sl(2). Adv. Math., 225:3327–3424, 2008. arXiv:0803.3652. [18] A.D. Lauda. An introduction to diagrammatic algebra and categorified quantum sl 2 . Bulletin Inst. Math. Academia Sinica, 7:165–270, 2012. arXiv:1106.2128. [19] A.D.Lauda, H.Queffelec, and D.Rose. Khovanov homology isaskew howe 2-representation of categorified quantum sl(m). arXiv:1212.6076. [20] G.Lusztig.Introduction to quantum groups, volume110ofProgress in Mathematics.Birkh¨ auserBostonInc.,Boston, MA, 1993. [21] Hoel Queffelec and David E. V. Rose. Sutured annular Khovanov-Rozansky homology. 2015. arXiv:1506.08188. [22] D.E.V. Rose. A Note on the Grothendieck Group of an Additive Category, 2011. arXiv:1109.2040. [23] R. Rouquier. 2-Kac-Moody algebras, 2008. arXiv:0812.5023. [24] B. Webster. Knot invariants and higher representation theory. 2013. arXiv:1309.3796. 80

Abstract (if available)

Abstract For a given quantum group $U_q(\frak{g})$, Lusztig defines a set of automorphisms $\tau_{i,e}'$, $\tau_{i,e}''$ on any integrable $U_q(\frak{g})$ module $M$. These automorphisms are categorified by certain Rickard Complexes which have played a tremendously important role in a number of recent results, most notably a proof of the Abelian Defect Conjecture. $U_q(\frak{g})$ itself is not an integrable $U_q(\frak{g})$-module, however, the action of $\tau_{i,e}$ on integrable modules is very closely related to a braid group action by automorphisms on the quantum group. These automorphisms satisfy the braid relations and are the key to constructing a PBW type basis for $U_q(\frak{g)}$. In this thesis we categorify these automorphisms and prove a compatibility condition relating them to Rickard complexes. Furthermore, we show that $\cal{T}_{i,e}$, our categorified versions of the $T_{i,e}$, also satisfy the braid relations. This categorification has potential implications for various ongoing projects in categorical representation theory.

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Creator Lamberto-Egan, Laffite M. (author)

Core Title A categorification of the internal braid group action of the simply laced quantum group

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 11/29/2016

Defense Date 10/03/2016

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

Tag braid group,categorification,OAI-PMH Harvest,quantum group

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Lauda, Aaron (committee chair), Friedlander, Eric (committee member), Montgomery, Susan (committee member), Zanardi, Paulo (committee member)

Creator Email llambert@usc.edu,piratelaffite1@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-324519

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