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Frobenius-Schur indicators for near-group and Haagerup-Izumi fusion categories
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Frobenius-Schur indicators for near-group and Haagerup-Izumi fusion categories
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FROBENIUS-SCHUR INDICATORS FOR NEAR-GROUP AND HAAGERUP-IZUMI FUSION CATEGORIES by Henry Tucker A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) May 2016 Copyright 2016 Henry Tucker To my parents, Nancy and Jay. ii Acknowledgments First I wish to extend my deepest gratitude to my dissertation adviser Susan Montgomery for her encouragement and support. I would not have found direction in my research without her guidance. Her advice and commentary were also invaluable to me during the production of this dissertation. I also want to thank Siu-Hung (Richard) Ng for help in beginning this work and for numerous useful and highly enjoyable conversations. Finally, I wish to thank my parents, Nancy and Jay, for all their love and encouragement. None of this would have been possible without their support. iii Table of Contents Dedication ii Acknowledgments iii Abstract v Chapter 1: Introduction 1 Chapter 2: Fusion categories 4 1 Basic denitions and diagrammatic calculus . . . . . . . . . . . . . . . . . . 4 2 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Quadratic fusion rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Chapter 3: Drinfel'd centers 15 1 Modular categories and modular data . . . . . . . . . . . . . . . . . . . . . 15 2 Frobenius-Schur indicators from the T -matrix . . . . . . . . . . . . . . . . . 19 Chapter 4: Quadratic forms and Gauss sums 21 1 Basic denitions and identities . . . . . . . . . . . . . . . . . . . . . . . . . 21 2 Evans-Gannon conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Chapter 5: Frobenius-Schur indicators 26 1 Type m =jGj 1 near-groups . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Type m =jGj near-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3 Haagerup-Izumi fusion categories . . . . . . . . . . . . . . . . . . . . . . . . 33 Chapter A: Categories of sectors 36 1 Murray-von Neumann subfactors . . . . . . . . . . . . . . . . . . . . . . . . 36 2 Sectors of Cuntz algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 Center data from the Ocneanu tube algebra . . . . . . . . . . . . . . . . . . 43 Bibliography 46 iv Abstract Ng and Schauenburg generalized higher Frobenius-Schur indicators to pivotal fusion cate- gories and showed that these indicators may be computed utilizing the modular data of the Drinfel'd center of the given category. We consider two classes of fusion categories gener- ated by a single non-invertible simple object: near groups, those fusion categories with one non-invertible object, and Haagerup-Izumi categories, those with one non-invertible object for every invertible object. Examples of both types arise as representations of nite or quantum groups or as Jones standard invariants of nite-depth Murray-von Neumann sub- factors. We utilize the Evans-Gannon computation of the tube algebras to obtain formulae for the Frobenius-Schur indicators of objects in both of these families. v Chapter 1 Introduction The representations of nite groups, Hopf algebras, and vertex operator algebras, invari- ants for topological quantum eld theories, knots, and Murray-von Neumann subfactors, and order parameters for topological states of matter are all examples of fusion categories. Therefore classication of fusion categories is in direct parallel with the classication pro- grams for all of these objects. The objects of fusion categories generalize the properties of complex representations of nite groups; objects are completely reducible into some irreducible objects, tensor products and duals are well-dened, and morphisms are linear maps. This point of view extends to the most important invariants of fusion categories, all of which were developed as generalizations from the representation theory of nite groups and Hopf algebras. These include categorical generalizations of the character ring, the quantum dimension, the classical and quantum 6j symbols, and the Frobenius-Schur indicator. Recall that this latter invariant was dened for representations of nite groups by the following formula for a representation V of nite group G where V is the corresponding character: 2 (V ) = 1 jGj X g2G V (g 2 ) The classical Frobenius-Schur theorem states that 2 (V )2f1;1; 0g and the value of(V ) determines whetherV is real, quaternionic, or complex. Already in the case of nite groups this invariant captures more information than the character ring. For example, consider the groupsD 8 the dihedral group of order 8 andQ 8 the quaternion group. It is easy to see that these have isomorphic character rings. However 2 (V ) = +1 for the degree 2 representation ofD 8 , but for the degree 2 representation of Q 8 we have 2 (V ) =1. Tambara and Yamagami considered this example in a categorical setting. Namely, they set out to prove that the categories of representations for the groupsQ 8 andD 8 are inequiv- alent as categories. This is a non-trivial result as it is indeed possible, as was shown by Etingof and Gelaki [EG], for two non-isomorphic groups to have equivalent categories of rep- resentations. To achieve this result Tambara and Yamagami classied all fusion categories having a certain fusion rule (which is the categorical generalization of the character ring) 1 by solving enormous systems of non-linear equations arising from the requirement that the tensor product be associative. They then showed that the two categories of representations above correspond to dierent categorical equivalence classes in their classication. The categorical Frobenius-Schur indicator provides a much simpler route to this result. Ng and Schauenburg provided this generalization of the indicator and showed that it is indeed a categorical invariant. Since the categorical indicator is equal to the classical indi- cator in the case of categories of complex group representations their theorem shows imme- diately that the categories of representations ofD 8 and Q 8 are not equivalent. Moreover, Ng and Shauenburg were also able to establish quasi-Hopf algebra realizations of the other equivalence classes of categories with the same fusion rule as the representations ofD 8 and Q 8 by using the categorical indicator. In fact, it was later shown by Basak and Johnson that the categorical Frobenius-Schur indicator provides a complete invariant for the equivalence classes of fusion categories having the Tambara-Yamagami fusion rules. This motivates the current work: do categories with fusion rules generalizing those of Tambara-Yamagami also possess this property? Izumi and Evans-Gannon have considered two related families of categories generated by a single object; one are the near-groups, those with only one non-invertible object, and the other are the Haagerup-Izumi fusion categories, those with one non-invertible object for each invertible object. The main examples of categories having these fusion rules appear as the Jones standard invariants of Murray-von Neumann subfactors. Izumi began a classication program for these categories by classifying the possible subfactors having these fusion rules. This method has proven to be very fruitful, particularly because it avoids the task of nding solutions to the systems of equations associated to the associativity rule. This realization also provides necessary data for the computation of the Frobenius-Schur indicators of these categories. Here we take advantage of their computations to obtain the indicators for these fusion categories, and in the process we establish new families of fusion categories whose equiva- lence classes are again completely determined by their indicators. We begin in Chapter 2 with the basic denitions for the theory of tensor categories, including the diagrammatic calculus utilized for understanding objects and morphisms in these categories and the three main invariants for distinguishing their equivalence classes. Here we also introduce the families of fusion categories whose invariants we will consider; these are fusion categories generated by a single object which are realized as Jones standard invariants for some families of Murray-von Neumann subfactors. 2 In Chapter 3 we consider fusion categories with the additional structure of a braiding and we dened the Drinfel'd center, which is the canonical braided category containing a given fusion category. The importance of this construction comes from the fact that the Frobenius-Schur indicators of the fusion category may be obtained from the braiding of their centers. In Chapter 4 we discuss the theory of quadratic forms on nite abelian groups. The center data for the known realized examples of singly-generated categories of present interest are given by such quadratic forms, and it is conjectured by Evans and Gannon that this is in fact always the case. In Chapter 5 we compute the Frobenius-Schur indicators for our families of singly-generated categories and show that they are given by Gauss sums, and, most importantly, we establish new families of fusion categories whose equivalence classes are completely determined by the Frobenius-Schur indicators. We also provide an Appendix outlining Izumi's method for classifying fusion categories by realization as standard invariants for subfactors. Using this method Izumi and Evans- Gannon provide classication parameters and the Drinfel'd center data that allow for our computations of the Frobenius-Schur indicators. 3 Chapter 2 Fusion categories In this chapter we will dene fusion categories and develop some basic usage of diagrammatic calculus. The main invariants utilized for the classication of fusion categories will also be introduced, and nally we will introduce the families of fusion categories under consideration presently. Much of the following are direct generalizations of classical results from the representation theory of nite groups. 1 Basic denitions and diagrammatic calculus All denitions and theorems in the following sections may be found in the monographs [EGNO] by Etingof, Nikshych, Gelaki, and Ostrik and [BaKi] by Bakalov and Kirillov. See also the textbook by Kassel [Kas] for a point of view from Hopf algebras. All categories herein will be taken to be abelian, that is direct sums of objects and morphisms and kernels and cokernels of morphisms all exist and possess some other natural desirable properties. See [Mac, Chapter VIII] for details. We begin with the following denition: Denition 2.1.1. A tensor category is an abelian categoryC equipped with: A bifunctor :CC!C called the tensor product, A natural isomorphism a : ( ) ! ( ) dening associativity for the tensor product and satisfying the pentagon axiom for all objects W;X;Y;Z2C: (W X) (Y Z) ((W X) Y ) Z W (X (Y Z)) (W (X Y )) Z W ((X Y ) Z) a W;X;Y Z a W X;Y;Z a W;X;Y id Z a W;X Y;Z id W a X;Y;Z 4 A unit object 12C and unit natural isomorphisms l : 1! Id C and r : 1 ! Id C satisfying the triangle axiom for all objects X;Y 2C: (X 1) Y X (1 Y ) X Y a X;1;Y l X id Y id X r Y It is clear that tensor categories are simply a categorication of the notion of a ring with identity, hence we also wish to consider the categorication of ring homomorphisms. This is accomplished by considering functors preserving the tensor structure: Denition 2.1.2. A tensor functor between tensor categories (C; C ) and (D; D ) is a functor F :C!D equipped with a natural isomorphism J :F () D F ()!F ( C ) satisfying the following coherence relation with the associativity structure: (F (X) D F(Y )) D F (Z) F (X) D (F (Y ) D F (Z)) F (X C Y ) D F (Z) F (X) D F (Y C Z) F ((X C Y ) C Z) F (X C (Y C Z)) F (a C X;Y;Z ) J X;Y id F(Z) id F(X) J Y;Z J X Y;Z J X;Y Z a D F(X);F(Y);F(Z) Finally, we have natural isomorphisms of tensor functors: Denition 2.1.3. A natural isomorphism of tensor functors is a natural isomorphism : (F;J F ) = (G;J G ) of tensor functors (C; C )! (D; D ) satisfying coherence with the tensor structure of the two functors for every object X;Y 2C: F (X) D F (Y ) F (X C Y ) G(X) D G(Y ) G(X C Y ) J F X;Y X D Y X C Y J G X;Y 5 Now we may dene an equivalence for tensor categories: Denition 2.1.4. Two tensor categories (C; C ) and (D; D ) are equivalent if there exists two tensor functors: (F;J F ) :C!D and (G;J G ) :D!C with natural isomorphisms GF = Id C and FG = Id D of tensor functors. A tensor category is called strict if the natural isomorphisms a, l, and r are identity maps for all objects of the category. Hence we may take (U V ) W andU (V W ) to be the same object in a strict tensor category. By the Mac Lane Strictness Theorem [Mac, Chapter XI,x3, Theorem 1] we know that any tensor category is equivalent to a strict tensor category. This allows us to employ diagrammatic calculus in a strict tensor category since there is no need to keep track of associativity in our graphical representations of objects and morphisms. Our diagrammatic notation is read from top to bottom, tensor products are given by side-by-side concatenation of object labels, the unit object 1 is not written at all, and morphisms are labeled strings whose compositions are denoted by stacking. For example, the morphisms id V :V !V; g :V !U W; f : 1!V 1 V n ; are rendered in diagrammatic notation, respectively, as: V V V g U W f V 1 V n Note that, because of strictness, we do not need parentheses for the domain object of the morphismf. Now we dene some other properties that a given tensor category may possess: Denition 2.1.5. A tensor categoryC is k-linear if it is enriched over the category of nite-dimensional vector spaces Vec fd (k); that is, Hom C (V;W )2 Vec fd (k) for all objects V;W2C. 6 Denition 2.1.6. A tensor categoryC is semisimple if every object is decomposable into a direct sum of some simple objects. The set of (isomorphism classes) of simple objects is denotedIrr(C). An object V in a k-linear tensor category is simple if and only if Hom C (V;V ) =k, and in fact we have the Schur property for simple objects V;W2Irr(C): Hom C (V;W ) = 8 < : k ifV =W 0 ifV W Denition 2.1.7. A tensor categoryC is rigid if every object V 2C has a corresponding (right) dual object V 2C with morphisms ev V :V V ! 1 and db V : 1!V V which are represented diagrammatically (assuming strictness) as V V and V V and satisfy the following relations: V V = V V and V V = V V We may now give the denition of the main object of interest: Denition 2.1.8. A fusion category is a k-linear, semisimple, rigid tensor categoryC such that 1 is simple andIrr(C) is nite. Examples of such tensor categories include: 1. Vec := Vec fd (k), the category of nite-dimensional vector spaces; this is the trivial fusion category with a single simple object (isomorphism class) 1 =k. 2. Vec G , the category of vector spaces graded by a nite group G. 3. Rep(G), the category of complex representations of a nite group G. 7 4. Rep(H), the category of complex representations of a semisimple (quasi-, weak) Hopf algebra H. 5. The principal even and dual even parts of the Jones standard invariant for a Murray- von Neumann subfactor N M of nite index and nite depth, denotedG + N;M and G N;M , respectively. (See Appendix,x1.) 2 Invariants The rst invariant for a fusion category is the ring structure of the tensor product. This is encoded via: Denition 2.2.1. The Grothendieck ring K 0 (C) for a fusion categoryC is theZ-based ring with basis the setIrr(C) and multiplication and addition given by and, respectively. Because we are working in a fusion category we may index the (isomorphism classes) of simple objectsIrr(C) =fX i g i2I by a nite set I and write the multiplication in K 0 (C) as: X i X j = X k2I N i;j k X k The set of coecientsfN i;j k := dim Hom(X i X j ;X k )g i;j;k2I are called the fusion rule, and the matrix (N i;j k ) j;k is called the fusion matrix for X i . This denition also allows us to establish a dimension for objects in a fusion category: Denition 2.2.2. The Frobenius-Perron dimension FPdim(V ) of an object V in a fusion categoryC is the maximum positive eigenvalue of left multiplication of V in the Grothendieck ring K 0 (C). In particular, FPdim(X i ) is the Frobenius-Perron eigenvalue for the fusion matrix of X i . Consider the Grothendieck ring for the aforementioned examples of fusion categories: 1. K 0 (Vec) =Z 2. K 0 (Vec G ) =Z[G] 3. K 0 (Rep(G)) =R(G), the character ring 4. K 0 (Rep(H)) =r(H), the Green ring 5. K 0 (G N;M ) = N;M , the principal graph (+) and the dual principal graph () 8 In examples 1-4 the FPdim corresponds to the usual linear dimension over the base eld, and in example 5 FPdim corresponds to the Murray-von Neumann dimension (see Appendix). Note that every fusion category such that FPdim(X)2Z for all simply objectsX2Irr(C) is realized as the category of representations of some quasi-Hopf algebra [ENO]. We may also see from the preceding that the Grothendieck ring is not a complete invariant for a fusion category. Consider the following important example: Example 2.2.3. LetC 1 = Rep(D 8 ), the complex representations of the dihedral group of order 8, and letC 2 = Rep(Q 8 ), the complex representations of the quaternion group. Now see that, for G =Z=(2)Z=(2), we have: K 0 (C 1 ) =K 0 (C 2 ) =Z[G[fg ] where multiplication is given by the group law and, for g2G: g = =g and 2 = X h2G h Hence the categoriesC 1 andC 2 are Grothendieck equivalent; we shall see later that they are, in fact, not equivalent as tensor categories! To see thatC 1 andC 2 are inequivalent as tensor categories we must appeal to the next important invariant: Denition 2.2.4. The 6j symbols for a fusion categoryC are the sum components of the linear maps induced from the associativity natural isomorphisms on triples of simple objects. That is to say, letIrr(C) = fX i g i2I and, given X i ;X j ;X k 2 Irr(C), consider the induced linear maps: (a X i ;X j ;X k ) : M n2I Hom((X i X j ) X k ;X n )! M n2I Hom(X i (X j X k );X n ) and consider the expansion via the fusion rule: (a X i ;X j ;X k ) : M n2I M m2I N i;j m N m;k n Hom(X n ;X n )! M n2I M l2I N i;l n N j;k l Hom(X n ;X n ) The name refers to the 6 indices required to address these maps: we look at the (l;m;n) component of the (i;j;k) triple of simple objects. 9 It is known that, given a fusion categoryC, the Grothendieck ring and the 6j symbols together completely determine C up to equivalence [EGNO, x4.9-4.10]. However, com- putation of 6j symbols is very unwieldy, often involving large systems of nonlinear equa- tions, hence they are of limited usefulness in classication of possible fusion categories for a given Grothendieck ring. Nonetheless, this method was successfully employed by Tam- bara and Yamagami [TY] in the classication of fusion categoriesC with Grothendieck ring K 0 (C) =Z[G[fg] with G a nite abelian group and multiplication given by the group law and, for g2G, g = =g and 2 = X h2G h This classication showed that a given Tambara-Yamagami category is determined up to tensor equivalence by the groupG (i.e., the Grothendieck ring) along with a non-degenerate bicharacterh;i onG and a complex number such that 2 =jGj 1 , these latter two param- eters giving the 6j symbols. Recall from Example 2.2.3 that the representation categories forD 8 and Q 8 are Tambara-Yamagami categories with G =Z=(2)Z=(2). Via the classi- cation in [TY] we see that = + 1 2 forD 8 and = 1 2 for Q 8 , hence the 6j symbols for the two categories are in fact inequivalent, and hence the two categories are inequivalent as fusion categories. This inequivalence can be seen much more readily by appealing to the Frobenius-Schur indicator, which is a ner invariant than the Grothendieck ring. This invariant was rst dened for complex representations of nite groups by Frobenius and Schur in 1906, and subsequently it was generalized to the settings of modules over semisimple Hopf algebras by Linchenko and Montgomery [LM], modules over semisimple quasi-Hopf algebras by Mason and Ng [MaN], and nally to objects of pivotal fusion categories by Ng and Schauenburg [NS07p]. Denition 2.2.5. A fusion categoryC is pivotal if it is equipped with a natural isomor- phism of tensor functors j : Id C ! () . It is conjectured by Etingof, Nikshych, and Ostrik [ENO] that every fusion category admits a pivotal structure. 10 Denition 2.2.6. LetC be a pivotal strict fusion category. Then, for k2N and an object V 2C, the k th Frobenius-Schur indicator k (V ) is given by the linear trace k (V ) = Tr 0 B B B B @ E (k) V : f V VV | {z } k 7! f j 1 V VV V 1 C C C C A of the rotation operator E (k) V , a linear endomorphism of the nite-dimensional vector space Hom( 1;V k ) where we take V k to be the k-fold tensor product of V with all parentheses to the right. It is shown in [NS07p] that the sequence of indicators for the simple objects is an invari- ant of the fusion category under equivalence. The categorical Frobenius-Schur indicator coincides with the classical version on categories of representations of nite groups, hence we may now use the indicators to establish inequivalence of the categories Rep(D 8 ) and Rep(Q 8 ). Specically, 2 () = 1 for Rep(D 8 ) and 2 () =1 for Rep(Q 8 ). More generally, 2 () =sign() for any Tambara-Yamagami category. These two group examples both have 4 () = 2; since the group G = Z=(2)Z=(2) has two inequivalent possible bicharacters there must be two other Tambara-Yamagami categories. These are those with 4 () = 0. One is Rep(K) where K is the Kac algebra of dimension 8, which is the last non-commutative semisimple Hopf algebra of dimension 8 by the classication of Masuoka [Mas], hence the fourth Tambara-Yamagami category is realized as representations of a quasi-Hopf algebra. Ng and Schauenburg construct a dimension 8 quasi-Hopf algebra in [NS08] by twisting the comultiplication on K by a non- trivial cocycle, and they establish its category of representations as a representative of this fourth equivalence class of Tambara-Yamagami categories again using the second Frobenius- Schur indicator. Furthermore, it was shown by Basak and Johnson [BasJ] that the Frobenius-Schur indicators completely determine the equivalence classes of Tambara-Yamagami categories, thus providing proof for a special case of the following conjecture: Conjecture 2.2.7. (S.-H. Ng & S. Montgomery) The equivalence classes of pivotal fusion categories tensor-generated by a single object are completely determined by their categorical Frobenius-Schur indicators. 11 The goal for the remainder of this work is to establish new families of singly-generated pivotal fusion categories satisfying this conjecture. This is primarily possible because this family of fusion categories is equipped with a particularly nice pivotal structure: Denition 2.2.8. A pivotal fusion categoryC is spherical if, for all V 2Irr(C), the left and right quantum (pivotal) dimensions agree: qdim r (V ) := j V V = V j 1 V =: qdim l (V ) In this setting the quantum dimension qdim(V ) := qdim r (V ) = qdim l (V ) of an object V 2 C is well-dened. Using this we may dene the quantum dimension of the fusion category: qdim(C) = X V2Irr(C) qdim(V ) 2 We should note that the above (right and left) quantum dimensions are special cases of the (right and left) quantum traces of morphisms f2 Hom(V;V ): qtr r (f) := f V j V and qtr l (f) := f V j 1 V where clearly qdim r (V ) = qtr r (id V ) and qdim l (V ) = qtr l (id V ). 3 Quadratic fusion rules Izumi and Evans-Gannon have classied some important families of singly-generated fusion categories by realizing them as systems of sectors associated to type III subfactors (see Appendix). These are the quadratic fusion categories. The rst important example of such fusion categories are those with only one non- invertible object: Denition 2.3.1. A near-group fusion categoryC is one with Grothendieck ring K 0 (C) =Z[G[fg] 12 whereG is a nite abelian group and where the multiplication inK 0 (C) is given by the group law and: g = =g forg2G and 2 =m + X h2G h where m is a non-negative integer. We call this a near group of type (G;m). The case where m = 0 has already been considered; these are the Tambara-Yamagami categories. Some other examples of near group fusion categories are: The categories of representations Rep(S 3 ) and Rep(A 4 ) for S 3 the symmetric group on three letters and A 4 the alternating group on four letters are near-groups of type (Z=(2); 1) and (Z=(3); 2), respectively. The principal even part of the D (1) 5 subfactor forms a near-group of type (Z=(2); 1). The principal even parts of the standard invariants for the A 4 , E 6 , and Izumi-Xu subfactors form near-groups of type (Z=(1); 1), (Z=(2); 2), and (Z=(3); 3), respectively. Evans-Gannon [EvGan12] prove a fundamental dichotomy in the possible values of the multiplicity m in terms of the order of G: Theorem 2.3.2. [EvGan12, Theorem 2(a)] The types for near-group fusion categories with group G are divided into two families: 1. m =jGj 1, or 2. m =kjGj for k2N[f0g On the other hand, Evans-Gannon have also considered singly-generated fusion cate- gories with one non-invertible object for each invertible object in G: Denition 2.3.3. A Haagerup-Izumi fusion categoryC is one with Irr(C) =G[fg g g2G where is a non-invertible, and where the Grothendieck ring is given by K 0 (C) =Z[G[fgg g2G ] with G a nite abelian group and where multiplication in K 0 (C) is given by the group law and: gh =hg 1 forg;h2G and gh =gh 1 + X x2G x 13 The most important examples of Haagerup-Izumi categories are the Jones standard invariant of the Haagerup subfactor, whereG =Z=(3), and the Yang-Lee system of sectors, which is the unique non-unitary Haagerup-Izumi fusion category with G the trivial group (see Appendix). These two families are the \extremal" examples of the following wider family of singly- generated fusion categories arising as standard invariants for subfactors: Denition 2.3.4. Let G be a nite abelian group, H G a subgroup, and G=H a set of (left) coset representatives. A quadratic fusion category is one with Irr(C) =G[f g g g2G=H where is a non-invertible object, and where the Grothendieck ring is given by K 0 (C) =Z[G[f gg g2G=H ] where multiplication in K 0 (C) is given by the group law and: =g for someg 2G; g = () g2g Hg 1 ; and 2 = X g2G=H n g g + X h2H h where n g are non-negative integers. Now we can see that the near-groups are quadratic with H = G, g = e G , and m = n e , and the Haagerup-Izumi fusion categories are quadratic with H =fe G g and n g = 1 for all g2G=H =G. 14 Chapter 3 Drinfel'd centers Here we study the braided setting. In particular, we will consider the canonical braided category containing a fusion category: the Drinfel'd center. Our interest arises here because the Frobenius-Schur indicators of spherical fusion categories may be obtained from the braiding of their Drinfel'd centers. 1 Modular categories and modular data Recall that some of our familiar examples of fusion categories come with a natural isomor- phism such that V W =W V for all objects V;W2C. For example: 1. In the category Rep(G) for a nite group G, we have: V W!W V v w7!w v This is a morphism ofCG-modules because G acts diagonally. 2. In the category Rep(H) for a quasi-triangular Hopf algebra (H;R) we have: V W!W V v w7!R 1 (w v) These are braided fusion categories. Denition 3.1.1. Let V op W := W V for V;W 2C, i.e. the tensor product in the opposite category. A fusion categoryC is braided if there exists a natural isomorphism c : ! op 15 satisfying the hexagon axioms for X;Y;Z2C: X (Y Z) (Y Z) X (X Y ) Z Y (Z X) (Y X) Z Y (X Z) c X;Y Z a Y;Z;X a X;Y;Z c X;Y id Z a Y;X;Z id Y c X;Z and (X Y ) Z Z (X Y ) X (Y Z) (Z X) Y X (Z Y ) (X Z) Y c X Y;Z a 1 Z;X;Y a 1 X;Y;Z id X c Y;Z a 1 X;Z;Y c X;Z id Y These natural isomorphisms are represented diagrammatically by: c V;W = V W WV and (c V;W ) 1 = WV V W and, whenC is strict, the hexagon axioms represent the following equalities of diagrams: XY Z Y Z X = X YZ YZ X and XY Z Z XY = XY Z Z XY where the left (resp. right) sides of the equalities can be seen by reading the bottom (resp. top) sides of the corresponding commutative diagrams from left to right. Note that in Example 2 for a quasi-triangular Hopf algebra the hexagon axiom is equivalent to the R-matrix satisfying the quantum Yang-Baxter equation. The braiding in the category Rep(G) is symmetric, i.e. c W;V c V;W = Id V W for all pairs V;W2C. That is, we have the following equality of diagrams: V W V W = V W V W 16 There is an opposite condition for braided fusion categories where the above equality implies one of the objects must be the unit object (and hence represented by the empty diagram). This condition is equivalent to the following: Denition 3.1.2. In a braided spherical fusion categoryC we dene the following matrix: S = [ qtr(c W;V c V;W ) ] V;W2Irr(C) = 0 B B B B B B B B @ V W 1 C C C C C C C C A V;W2Irr(C) The braiding is called non-degenerate if the S-matrix is invertible. A modular category is a non-degenerately braided spherical fusion category. In a braided fusion category pivotal structures are in one-to-one correspondence with ribbon structures. A ribbon structure is a natural automorphism : Id C ! Id C satisfying the following identity for objects V;W2C: V W = ( V W )c W;V c V;W We may obtain the ribbon structure associated with our pivotal structure j by using the Drinfel'd isomorphism u : Id C ! () , which is given diagrammatically as: u V = V V 17 Then we have a ribbon structure given by = j 1 u where the ribbon axiom is given diagrammatically as: VW j 1 VW VW = V W j 1 V j 1 W V W Now we have a second matrix invariant for modular categories: Denition 3.1.3. LetC be a modular category. This category has a ribbon structure given by =j 1 u, and we dene the following diagonal matrix: T = Diag 0 B B B B B B B @ V = V j 1 V V 1 C C C C C C C A V2Irr(C) The matrices S and T are called the modular data for the modular categoryC. The reason for this is that modular data for a modular category give a projective representation of the modular group SL 2 (Z) in the following way: s = 0 1 1 0 ! 7! 1 p qdimC S and t = 1 1 0 1 ! 7!T where s and t are the generating matrices for SL 2 (Z). This also gives us the following identities: (ST ) 3 =S 2 X V2Irr(C) V qdim(V ) 2 and S 4 = qdim(C) 2 I Given a spherical fusion categoryC we may construct a braided spherical fusion category containing it: Denition 3.1.4. Let C be a spherical fusion category. The Drinfel'd center is the categoryZ(C) whose objects are pairs (V;e V ) of objects V 2C and half-braiding natural isomorphisms e V :V ! V 18 such that e V X Y = (id X e V Y ) (e V X id Y ) This construction yields a braided spherical fusion category. The coherence axiom in the preceding denition is represented graphically as the following equality of diagrams: V XY XY V = V XY XY V Hence it is equivalent to requiring that the half-braiding e V X must satisfy the rst hexagon axiom for every object X2C. It was shown by M uger in [M u] that the Drinfel'd center is in fact a modular category. An important example of the Drinfel'd center is given whenC = Rep(H) for a spherical semisimple Hopf algebra: Z(Rep(H))' Rep(D(H)) where D(H) is the quantum double of the Hopf algebra (cf. [M, Chapter 10]). 2 Frobenius-Schur indicators from the T-matrix Ng and Schauenburg [NS07s] have shown that one may obtain the Frobenius-Schur indica- tors for objects in a fusion categoryC from the modular data of its Drinfel'd centerZ(C). Let F :Z(C)!C be the forgetful functor. By a result of M uger [M u] this functor has an adjoint functor I :C!Z(C) called the induction functor such that: I(V ) = M (X;e X )2Irr(Z(C)) (X;e X ) dim Hom C (V;X) where dim is the usual linear dimension. Now we have the following theorem which produces the Frobenius-Schur indicators from the entries of the T -matrix: Theorem 3.2.1. [NS07s, Theorem 4.1] LetC be a spherical fusion category and let V 2C. Then: k (V ) = 1 qdim(C) qtr k I(V ) 19 Combining this theorem with the identication above we have the following formula coming directly from the T -matrix ofZ(C): k (V ) = 1 qdim(C) X (X;e X )2Irr(Z(C)) k (X;e X ) qdim(X) dim Hom C (V;X) 20 Chapter 4 Quadratic forms and Gauss sums It was shown by Basak and Johnson [BasJ] that the categorical Frobenius-Schur indicators for the Tambara-Yamagami categories are given by quadratic Gauss sums associated to the bicharacter from the classication in [TY]. Using this fact and Wall's classication of quadratic forms on nite abelian groups [Wa] they were able to show that the tensor equivalence classes of Tambara-Yamagami categories are in fact completely determined by their categorical Frobenius-Schur indicators. In this chapter we provide some necessary background on the theory of quadratic forms on nite abelian groups. We also discuss two conjectures of Evans and Gannon on the modular data of near-groups and Haagerup-Izumi fusion categories; it is their prediction that the S and T matrices for these families are always given by quadratic forms. 1 Basic denitions and identities Let G be a nite abelian group; a complex symmetric bicharacter h;i :GG!C is simply a function multiplicative in both variables such thathg;hi =hh;gi for allg;h2G. Such a bicharacter determines a homomorphism between the group G and its Pontryagin dual group b G: h;i \ :G! b G g7!h;gi The kernel of this map is called the radical ofh;i: Radh;i :=kerh;i \ =fh2Gjhg;hi = 1 for allg2Gg 21 The bicharacter is called non-degenerate ifRadh;i is trivial. Bicharacters are in correspon- dence with bilinear forms b :GG!Q=Z on the group via the exponential function: hg;hi =e 2ib(g;h) The concepts of symmetry and non-degeneracy translate accordingly. From this identica- tion we consider quadratic forms associated with a given bicharacter. Denition 4.1.1. A function q : G!Q=Z is a quadratic form if q(g) = q(g) for all g2G and the associated bilinear form @q :GG!Q=Z given by @q(g;h) =q(gh)q(g)q(h) is symmetric. The quadratic form is called non-degenerate if the associated bilinear form is non-degenerate. A pair (G;q) is called a pre-metric group. If the bilinear form @q is non-degenerate then the pair is called a metric group. We may now dene an important invariant of quadratic forms on nite abelian groups: Denition 4.1.2. Let (G;q) be a pre-metric group. Then the associated Gauss sum is given by: (G;q) = 1 p jGj X g2G e 2iq(g) We are interested in the Gauss sums for quadratic forms q such thathg;hi =e 2i@q(g;h) for the non-degenerate bicharactersh;i appearing in the classications of near-group fusion categories. The Tambara-Yamagami categories have already been considered by Basak- Johnson [BasJ] in this direction. They showed the following result: Theorem 4.1.3. [BasJ, Lemma 4.1] LetC be a Tambara-Yamagami category with classi- cation parameters (G;h;i;) and let q be a quadratic form associated toh;i. Then, for integers k 1, we have: 2k () =sign() k (G;q) k (G;kq) 2k1 () = 0 and these values do not depend on the choice of q. 22 If the groupG is odd order then the correspondence between non-degenerate symmetric bicharacters and quadratic forms is a bijection, hence in the odd order case one need not worry about what quadratic form is chosen for computing the Gauss sums for the above indicators. More sophisticated results from analytic number theory are required to handle the even order case; this was achieved in the above result by appealing to a generalization of the Kervaire invariant for quadratic forms. We will restrict ourselves to the case where G is odd order for the families of fusion categories under consideration here; the even case will be considered in future work. If (G 1 ;q 1 ) and (G 2 ;q 2 ) are two pre-metric groups we denote the pre-metric group given by their orthogonal direct sum by (G 1 ;q 1 )? (G 2 ;q 2 ). This is the pre-metric groupG 1 G 2 with quadratic form q 1 +q 2 dened by: (q 1 +q 2 )(g 1 ;g 2 ) :=q 1 (g 1 ) +q 2 (g 2 ) It is easy to see that the Gauss sum is multiplicative under this construction: ((G 1 ;q 1 )? (G 2 ;q 2 )) = (G 1 ;q 1 )(G 2 ;q 2 ) Therefore understanding the Gauss sums of a general nite abelian group reduces to under- standing the Gauss sums of the metric groups irreducible under the orthogonal direct sum. For odd-order groups these are the following, as determined by Wall in [Wa]: label for (G;q) group G form q A p r Z=(p r ) q A p r (g) = g 2 p r B p r Z=(p r ) q B p r (g) = upg 2 p r where u p is a quadratic non-residue modulo p. Now we may utilize the following pair of lemmas to compute Gauss sums from the orthogonal direct sum decomposition of a given odd-order nite abelian group: Lemma 4.1.4. [BasJ, Lemma 3.2(a)] Let p be an odd prime and let k be an integer with gcd(k;p) = 1. Then: Z=(p r ); kg 2 p r = k p r p r where ( k p ) is the Legendre symbol and n = 8 < : 1 ifn 1 mod 4 i ifn 3 mod 4 23 Lemma 4.1.5. [BasJ, Lemma 3.3(a)] Again let p be an odd prime and let q be a non- degenerate quadratic form on the group G = (Z=(p r )) n . Then p s q induces a quadratic form on G=p rs G and: (G;p s q) =p sn=2 (G=p rs G;p s q) Utilizing these two lemmas Basak-Johnson showed that if two Tambara-Yamagami cate- gories are inequivalent then the sequences of Frobenius-Schur indicators are inequivalent. We will do the same for some families of near-groups in the sequel. 2 Evans-Gannon conjectures Evans-Gannon have established the following two conjectures in their classications of near- groups and Haagerup-Izumi categories and their subsequent computations of their centers' modular data. Specically, they conjecture that the modular data for the centers of near- groups are always given by quadratic forms. Recall that F :Z(C)!C is the forgetful functor. Conjecture 4.2.1. [EvGan12, Conjecture 2] Suppose thatC is a near-group fusion cate- gory of type (G;jGj) wherejGj is odd andh;i is the bicharacter from the Evans-Gannon classication parameters. Then there exists a metric group (G 0 ;q 0 ) of orderjGj + 4 such that the simple objects fEjF (E) =gIrr(Z(C)) are indexed by g2G;x2G 0 nfeg where E g;x =E g;x 1 and: Eg;x =hg;gie 2i@q 0 (x) In fact, Evans-Gannon establish formulae for obtaining S andT matrices associated to any pair of metric groups (G;q) and (G 0 ;q 0 ); the above conjecture simply states that near-group center modular data is given by these formulae. They provide the following criterion for such S and T matrices produced from their formulae to be modular data: Proposition 4.2.2. [EvGan12, Proposition 7(b)] The pair of metric groups (G;q) and (G 0 ;q 0 ) give modular data if and only if (G;q)(G 0 ;q 0 ) =1 24 Evans-Gannon have shown in [EvGan12] that the conjecture is true for near-group fusion categories withjGj 13 odd. They provide a similar result for the Haagerup-Izumi family: Conjecture 4.2.3. [EvGan15, Conjecture 1] SupposeC is a Haagerup-Izumi fusion cate- gory associated to the nite abelian group G withjGj odd. Then there exists a metric group (H;q 00 ) of orderjGj 2 + 4 = 2m + 1 such that the simple objects 8 < : D F (D) = M g2G g 9 = ; Irr(Z(C)) are indexed by h2Hnfeg where D h =D h 1 and: D h =e 2imq 00 (h) Evans-Gannon have shown in [EvGan15] that the conjecture is true for all Haagerup-Izumi fusion categories withjGj 5 odd. 25 Chapter 5 Frobenius-Schur indicators In this chapter we will produce formulae for the Frobenius-Schur indicators of the generating non-invertible object for the families of fusion categories under consideration thus far. We will also show that the sequences of indicators provide a complete invariant for some of the near-group families. 1 Type m =jGj1 near-groups LetC be a near-group fusion category of type with group of invertible simple objectsG with m =jGj 1. By [EvGan12, Proposition 2] the only non-empty tensor equivalence classes of near groups of this type are those whereG =F jGj+1 . SoG is cyclic, and thusH 2 (G;T) = 1. Let p = char(F jGj+1 ). Evans and Gannon showed that the tensor equivalence classes for fusion categories having this Grothendieck ring with group G are in correspondence with solutions to a system of equations in some parameters s2f1g and a;b;b 00 2 T (see Appendix). The tensor equivalence class for G given by Rep(AGL 1 (F jGj+1 )) corresponds to the solution s =a =b =b 00 = 1 in the Evans-Gannon classication parameters, and these are the only equivalence classes of this type unlessjGj = 1; 2; 3; 7. 1.1 Indicators forC' Rep(AGL 1 (F q )) We may use classical methods to determine the indicators forC that is tensor equivalent to the category of representations of an ane general linear group of degree 1 over the nite eldF q . Recall that G k (h) =jfg2Gjg k =hgj. Proposition 5.1.1. SupposeC is a near-group fusion category of type (G;jGj 1) and jGj6= 1; 2; 3; 7. ThenC' Rep(AGL 1 (F jGj+1 )) and: k () = G k (e) 1 + b k p c; k p 26 Proof. LetjGj + 1 =q. SinceAGL 1 (F q ) =F q oF q we may use [Se,x8.2, Proposition 25] to see that the character for the non-tensor inveritlbe irreducible representation is given by: (a;b) = 1;b q X (x;y)2AGL 1 (Fq ) (y 1 a) for any non-trivial linear character 2 \ (F q ; +). Now we may apply the classical formula for k () [I, Lemma 4.4]: k () = 1 q(q 1) X (a;b)2FqoF q ((a;b) k ) = 1 q(q 1) X (a;b);b k =1 ((1 +b +b 2 + +b k )a; 1) = 1 q(q 1) 0 @ X (a;b);b k =1;b6=1 (0; 1) + X n2Fq (kn; 1) 1 A Since is a degree q 1 character, the left hand sum is q(q 1)( F q k (1) 1). The right hand sum is equal to: X n2Fq X b2F q (b 1 kn) = 8 < : q(q 1) if pjk 0 if p-k The pjk case is clear since then (b 1 kn) is identically 1. On the other hand, (b 1 kn) = b(kn) under the transpose of the left regular action ofF q =GL 1 (F q ) on c F q =F q . Since (p;k) = 1 we have that: X n2Fq b(kn) = X n2Fq b(n) and since the action is faithful by denition, we know thatb is not the trivial representation for any b2 F q . Hence by orthogonality of characters the sum is 0. The formula is now clear since the given Kronecker delta is 1 if pjk and is 0 otherwise. 27 1.2 Indicators for via T-matrix ofZ(C) ForjGj = 1; 3; 7 there is 1 additional tensor equivalence class, and forjGj = 2 there are 2 additional tensor equivalence classes. Using the modular data computed in [EvGan12, Theorem 5] we will appeal to Theorem 3.2.1 to compute their indicators in general. Excluding the case wherejGj = 7 and s =1 we have the following data forZ(C): Simple object family X2Irr(Z(C)) F (X) Half-braid parameters X A g (g2G) g 1 1 M x2G x 1 1 B g (g2G) +g 2 b Gnfg (g) C 2 \ F + jGj+1 1 (1) where the half-braiding for C on occurrences of in objects ofZ(C) is a morphism: e C ()2 Hom( 2 ; 2 ) =C jGj M m (C) given by: e C () = 1 (1) M k2G (1) mk Id k ! M ( )( ) 2 ( ) ; Id ; For the case wherejGj = 7 and s =1 we have: Simple object family X2Irr(Z(C)) F (X) Half-braid parameters X A g (g2G) g 1 1 M x2G x 1 1 B g (g2G) +g 2 b Gnfg (g) E 1 + 1 i E 2 + 1 i 28 With the preceding data in hand we may now apply Theorem 3.2.1 to see: Theorem 5.1.2. SupposeC is a near-group fusion category of type (G;jGj 1). Then the indicators for the non-invertible object are given by: 1. IfjGj6= 7 or s = 1 then: k () = ( G k (e) 1) + 1 k b k p c; k p 2. IfjGj = 7 and s =1 then: k () = ( G k (e) 1) + (1) k=2 b k 2 c; k 2 Proof. (1) SupposejGj6= 7 or s = 1. Then we have: k () = 1 FPdimC 0 B B B @ X g2G !2 b Gnfg k B g FPdim(B g ) + X 2 \ F + jGj+1 k C FPdim(C ) 1 C C C A = 1 FPdim(C) 0 B B B @ (jGj + 1) X g2G 2 b Gnfg (g) k +jGj 1 k X 2 \ F + jGj+1 (1) k 1 C C C A Consider the rst summand. SinceG is abelian we may choose an isomorphismh7! h from G! b G. Then we have: X g2G 2 b Gnfg (g) k = 0 @ X g2G X 2 b G (g) k 1 A 0 @ X g2G (g) k 1 A = 0 @ X g2G X h2G h (g) k 1 A jGj = jGj X h2G k ( h ) ! jGj =jGj( G k (e) 1) 29 Consider the second summand. SinceF + n+1 is the additive group of a nite eld we have thatn + 1 =p l for some primep and positive integerl and thatF + n+1 = (Z=(p)) l as groups. Under this identication the multiplicative unit 12F + n+1 is a direct sum of generators of the copies ofZ=(p). X 2 \ F + jGj+1 (1) k = X 2 \ F + jGj+1 (k1) = 0 if k16= 0 p l if k1 = 0 = 0 if p-k jGj + 1 if pjk = (jGj + 1) b k p c; k p (2) Now suppose thatjGj = 7 and s =1. Then: k () = 1 FPdimC 0 B B B @ X g2G !2 b Gnfg k B ! g FPdim(B ! g ) + 2 2 X i=1 k Et FPdim(E t ) 1 C C C A = G k (e) 1 + 4jGji k (1 + (1) k ) jGj +jGj 2 = G k (e) 1 + i k (1 + (1) k ) 2 = G k (e) 1 + (1) k=2 b k 2 c; k 2 Corollary 5.1.3. Near-group fusion categories of type (G;jGj 1) can be completely dis- tinguished by their categorical Frobenius-Schur indicators. Proof. The statement is vacuous in all but the cases wherejGj = 1; 2; 3; 7. We shall consider them now: IfjGj = 1; 3; 7 then there is one additional tensor equivalence class corresponding to s =1. By [EvGan12, p.41] ifjGj + 1 is even (i.e. a power of 2) then 2 1 = s, hence 2 () =s in each of these three cases. IfjGj = 2 then s = 1 but instead b = ! where ! is some third root of unity. The two 30 non-trivial possibilities for! correspond to the two additional tensor equivalence classes for this type. By [EvGan12, p.42] if ! =e 2i 3 then 1 =e 2i 3 , hence 3 () =!. 2 Type m =jGj near-groups The modular data for the centerZ(C) ofC a near-group with group of invertible objects G and m =jGj was computed by Izumi in [Iz00ii, Theorem 6.8] and is given as follows: Simple object family X2Irr(Z(C)) F (X) Half-braid parameters X A g (g2G) g 1 hg;gi B g (g2G) +g 1 hg;gi C g;h =C h;g (g;h2G) +g +h 1 hg;hi E j jGj(jGj+3) 2 ! j The ! j 2 1 T are solutions to the system of equations (6.18)-(6.20) in [Iz00ii,x6] parametrized by g2G with coecients given by the complex parameters a(g);b(g);c2C in Izumi's classication of these categories (see Appendix). Proposition 5.2.1. SupposeC is a near group category of type (G;jGj) with non-invertible object and classication bicharacterh;i. Let q be a quadratic form such thathg;hi = e 2i@q(g;h) . Then the indicators for are given by the following: k () = 1 2 G k (e) + qdim() qdim(C) 0 @ p jGj 2 (G; 2kq) + jGj(jGj+3)=2 X j=1 ! k j 1 A Proof. Let d := qdim() and let < be an arbitary ordering on the nite group G. Again applying Theorem 3.2.1 we have the following: k () = 1 qdim(C) 0 B B @ (1 +d ) X g2G k Bg + (2 +d ) X g;h2G g<h k C g;h +d jGj(jGj+3)=2 X j=1 k E j 1 C C A = 1 qdim(C) 0 @ d 2 X g2G hg;gi k + 2 +d 2 X g;h2G hg;hi k +d X j ! k j 1 A 31 where the second equality is due to the symmetry of . Now we consider the middle sum: X g;h2G hg;hi k = X g2G X h2G hg;h k i =jGj X g2G groups k (hg;i) =jGj G k (e) The second equality is by denition of the Frobenius-Schur indicator for nite groups (denoted groups k ) [I, Equation 4.4] and the third equality is by [I, p. 49]. Now let q be a quadratic form such thathg;hi =e 2i@q(g;h) and consider the rst sum: X g2G hg;gi k = X g2G e 2i(2kq(g)) = p jGj(G; 2kq) Hence the formula is now clear. 2.1 Indicators for whenjGj is odd WhenjGj is odd we have that the correspondence between quadratic forms and bilinear forms given by the map q7! @q is a bijection. So let q be the quadratic form on G such thathg;hi =e 2i@q(g;h) . Theorem 5.2.2. Suppose Conjecture 4.2.1. Then: k () = 1 2 G k (e) + 1 2 (G; 2kq)(G 0 ; 2kq 0 ) Proof. Let N = (jG 0 j 1)=2 and enumerate G 0 as follows: G 0 =fe;x 1 ;:::;x N ;x 1 1 ;:::;x 1 N g 32 Let d :=qdim() andhx;yi 0 :=e 2i@q 0 (x;y) . Starting with Proposition 5.2.1 we have: k () = 1 2 G k (e) + d qdim(C) 0 B B @ p jGj 2 (G; 2kq) + X g2G 1iN hg;gi k (hx i ;x i i 0 ) k 1 C C A = 1 2 G k (e) + d p jGj(G; 2kq) jGj(2 +d ) 0 @ 1 2 + X 1iN (hx i ;x i i 0 ) k 1 A = 1 2 G k (e) + d p jGj(G; 2kq) jGj(2 +d ) 1 2 + 1 2 ((G 0 ; 2kq 0 ) p jGj + 4 1) = 1 2 G k (e) + d p jGj p jGj + 4 2jGj(2 +d ) (G; 2kq)(G; 2kq 0 ) and using the fact that d 2 =jGj +jGjd we have: d p jGj p jGj + 4 2jGj(2 +d ) = d (2d jGj) 2jGj(2 +d +) = 1 2 and then the formula for k () is clear. Combining the preceding theorem with Proposition 4.2.2 we have the following corollary: Corollary 5.2.3. (G;q) and (G 0 ;q 0 ) give realizable modular data via the formulae of Evans- Gannon if and only if 1 () = 0. This is an obvious requirement as the rst Frobenius-Schur indicator for a non-unit simple object must be zero. 3 Haagerup-Izumi fusion categories WhenC is a Haagerup-Izumi fusion category with group of invertible objectsG the modular data for the centerZ(C) was computed by Evans{Gannon in [EvGan15,x6.3] and is given as follows: 33 Simple object family X2Irr(Z(C)) F (X) # Half-braid parameters X 1 1 1 1 B 1 + P g2G g 1 1 A =A 21 + P g2G g 2 b G 1 C (h) =C (h 1 ) (h2G) h+h 1 + P g2G g 2 b G (g) D j P g2G g 1j jGj 2 +3 2 j The j are a solutions to a system of equations given in [EvGan15,x6.2]. Now we have: Theorem 5.3.1. SupposeC is a Haagerup-Izumi fusion category withjGj odd satisfying Conjecture 4.2.3, that is the modular data for the simple objects D j are given by a metric group (H;q 00 ) wherejHj =jGj 2 + 4 = 2m + 1. Then: k () = 1 2 G k (e) + 1 2 (H;kmq 00 ) Proof. Let d =qdim() in the categoryC. Again by using Theorem 3.2.1;: k () = 1 qdim(C) qdim(B) + X 6= 2 b G qdim(A ) + X e6=h 1 6=h2G X 2 b G k C h qdim(C h ) + X 1 6= 2H D qdim(D ) LettingjGj = 2n + 1 andjHj = 2m + 1 we may enumerate these odd order groups as: G =fe; g i ; g 1 i j 1ing and H =fe; h j ; h 1 j j 1jmg this gives us: k () = 1 qdim(C) jGj +jGjd +jGjdn + (2 +jGjd) X g i ; (g i ) k +jGjd X h j 2H k j 34 By the same argument as in the proofs of Theorems 5.1.2 and 5.2.2 we can see: X g i ; (g i ) k = jGj 2 ( G k (e) 1) Hence using the expression for the center's ribbon structure from Conjecture 4.2.3 and the fact thatjHj =jGj 2 + 4 and qdim(C) = 2jGj +djGj 2 we see: k () = jGj qdim(C) 2 +jGjd 2 G k (e) +d +dn jGjd 2 +d X h j 2H e 2ikmq 00 (h j ) = 1 2 G k (e) + jGj 2qdim(C) 2d + 2dnjGjd +d( p jHj(H;kmq 00 ) 1) = 1 2 G k (e) + jGjd p jGj 2 + 4 2qdim(C) (H;kmq 00 ) = 1 2 G k (e) + 1 2 (H;kmq 00 ) 35 Appendix A Categories of sectors In this appendix we describe the operator algebraic methods initiated by Izumi used for the classication of near-group fusion categories by both him and Evans-Gannon. Similar methods were used by Evans-Gannon in the case of the Haagerup-Izumi fusion categories, but these fusion categories are not always realized as unitary fusion categories and thus require some technical details to generalize the methods presented here. Nevertheless, the process of obtaining the modular data of their centers from the Ocneanu tube algebra is largely the same as in the unitary case; however, for the sake of brevity we only address the case of near-groups here. The representation theory for a nite-index, nite-depth Murray-von Neumann subfac- tor N M is given by a pair of fusion categories: the Jones standard invariantG N;M . Thus, the classication programs of subfactors and fusion categories are in parallel. With this in mind, Izumi and Evans-Gannon classied the near-group fusion categories by realiz- ing them as standard invariants of some subfactors. This is accomplished by appealing to Longo's axiomatization [Lo] of the standard invariant as a system of sectors, i.e. unitary equivalence classes of endomorphisms of a von Neumann algebra. A fusion category of sectors is convenient to work with as the tensor product is simply given by composition of endomorphisms, and one may avoid using 6j symbols in classifying fusion categories with a given Grothendieck ring. 1 Murray-von Neumann subfactors All of the following denitions and theorems and the required background on Hilbert spaces and C algebras may be found in the monograph by Evans and Kawahigashi [EvKa]. The monograph by Bischo, Kawahigashi, Longo, and Rehren [BKLR] and Izumi's papers [Iz91] and [Iz00i] contain all the facts we use about the theory of tensor categories of sectors. LetB(H) be the algebra of bounded operators on a separable Hilbert space H. This is a-algebra where for a2B(H) we denote the adjoint a which satises h 1 ;a ( 2 )i H =ha( 1 ); 2 i H for all 1 ; 2 2H 36 Let MB(H) be a-subalgebra; we dene the commutant of M by: M 0 =fa2B(H)jax =xa for allx2Mg The following denition is a consequence of the von Neumann double-commutant theorem: Denition A.1.1. A-subalgebra MB(H) is a von Neumann algebra if it has one of the following two equivalent properties: 1. M =M 00 2. M is closed in the weak (operator) topology A von Neumann algebra M is called a factor if Z(M) =M\M 0 =C1 M We contrast this with the denition of a C algebra, which (as a consequence of the Gelfand-Naimark theorem) is dened to be a-subalgebra ofB(H) that is closed in the norm topology, which is a ner topology than the weak topology. By the work of Murray and von Neumann every von Neumann algebra is a direct integral of factors, which is a continuous analog of a direct sum. Factors are algebraically simple, hence the study of their morphisms is equivalent to the study of unital inclusions of factors, i.e. subfactors. This is analagous to the situation of elds and eld extensions. Moreover, as we shall see, this analogy may be extended via generalizations of the degree and the Galois group due to Jones [J]. Factors come in three types based upon their lattice of projections, i.e. elementsp2M such that p 2 = p = p . These are exactly the operators whose images are orthogonal projections onto closed subspaces of H. This establishes a partial order on the projections of M where p q () p(M) q(M), and we call two projections p;q2 M Murray-von Neumann equivalent if there exists a partial isometryu2M such thatp =uu andq =u u, i.e. if their images can be mapped isometrically onto each other. Let M be a factor; we have the following classication due to Murray and von Neumann: Type I If M has a non-zero minimal projection: a projection p such that there is no non-zero projection q2M with qp, i.e. there are no proper sub-projections. Type II IfM has no non-zero minimal projection, but there exists non-zero nite projec- tions, i.e. a projection p2 M such that there exists no proper sub-projection q p such that q is Murray-von Neumann equivalent to p. Type III If M has no non-zero nite projections. 37 Factors of type I are simply those isomorphic toB(H) for some Hilbert space H, hence these are simply matrix rings in the nite dimensional case. We will now use the type II setting to develop the Jones theory of subfactor invariants. The most important property of type II factors M is that they are equipped with a trace functional tr M : M! [0;1]. We specialize to the type II 1 case, which are those factors where we may normalize the trace so that tr M (1 M ) = 1. (Equivalently, these are type II factors where 1 M is a nite projection.) The existence of this trace allows us to consider the left regular representation of M on L 2 (M;tr M ) =: L 2 (M), the Hilbert space obtained by completion of M in the normjjvjj :=tr M (v v) 1=2 . (This yields a faithful representation as an application of the Gelfand-Naimark-Segal construction.) This is the fundamental example of a (left) Hilbert space module for the von Neumann algebraM. We now have the following denitions: Denition A.1.2. Let N M be an inclusion of type II 1 factors. The Jones index [M :N] is dened as: [M :N] := dim N (L 2 (M)) where dim N is the Murray-von Neumann dimension or coupling constant for Hilbert space N-modules, cf. [EvKa, Denition 5.31]. We say that two Hilbert space bimodules H and K (i.e. Hilbert space modules with both a left and right action) over a von Neumann algebraM are unitarily equivalent if there exists unitary u2 M such that H = uKu . (This is clearly a stronger notion than the usual Murray-von Neumann equivalence.) Denition A.1.3. If [M :N]<1 then the standard invariantG N;M is the pair of tensor categories of (unitary equivalence classes of) Hilbert space bimodules under the Connes relative tensor product (cf. [EvKa, Denition 9.61]) generated by L 2 (M) considered as a bimodule in the following two ways: the principal even partG + N;M , the tensor category of NN bimodules tensor- generated by L 2 (M), and the dual even partG N;M , the tensor category of MM bimodules tensor-generated by L 2 (M). If there are only nitely many simpleNN bimodules then these are fusion categories, and we say that MN is nite-depth. 38 A type II 1 factor is called hypernite if it is generated by an increasing union of nite- dimensional algebras. By deep results of Popa [P] the standard invariant is a complete invariant for nite-index, nite-depth hypernite II 1 subfactors. Now we consider the type III setting, which poses the following diculty: for a type III factor M any two Hilbert space modules H and K are unitarily equivalent. However, by the work of Longo [Lo], we may obtain the standard invariant for type III subfactors by considering instead the algebra of endomorphisms of M. The equivalence of these two approaches may be seen as follows. Again consider a type II 1 factor M with an endomor- phism :M!M. Then we may dene an MM bimodule M isomorphic to L 2 (M) as a Hilbert space with actions of M given by: xy :=x(y) forx;y2M;2L 2 (M) Not all bimodules for type II 1 factors arise in this way, however they do in the type III case. Furthermore, under the Connes relative tensor product we have M M = M , hence in the type III case the standard invariant is given as a tensor category of (unitary equivalence classes of) endomorphisms with tensor product given by composition. In this category the direct sum is dened using adjoint representations of subprojections of 1 M (recall this is type III) and the dual of an endomorphism is given by composition with the Longo canonical endomorphism of the subfactor (M) M. See [Lo], [Iz91], and [Iz00i, x2] for details on these operations. Popa's results in [P] extend to the case of hypernite type III subfactors, hence again nite-index, nite-depth subfactors of the hypernite type III factor are completely deter- mined by their standard invariants realized as unitary equivalence classes of endomorphisms. This axiomatization of the standard invariant for type III subfactors has proven to be very computationally ecient for the purpose of classifying all fusion categories associated to a given Grothendieck ring. 2 Sectors of Cuntz algebras An important consequence of the work of Popa is that every C fusion category, i.e. fusion categories whose simple objects' endomorphism spaces are equipped with a C -algebra structure, may be realized as the quotient under unitary equivalence of a subcategory of End 0 (M) :=f2 End(M)j [M :(M)]<1g 39 for the hypernite type III factor M. These are fusion categories of sectors on M, i.e. unitary equivalence classes of endo- morphisms in End 0 (M). Two endomorphisms ;2 End 0 (M) are unitarily equivalent if and only if: = Ad u for some unitaryu2M The adjoint action Ad : M! End 0 (M) of unitaries on M is given by Ad u (x) = uxu for x2 M. The category of such equivalence classes is denoted Sect(M). In this category the tensor product is given by composition and the Hom spaces are given by spaces of intertwiners: Hom(;) =fv2Mjv(x) =(x)v for allx2Mg Both Izumi [Iz00ii][Iz15] and Evans-Gannon [EvGan12] produce classications of near- group fusion categories by realizing them as a system of sectors on the type III factor given by the weak closure of the Cuntz algebraO n . This is the C -algebra generated by isometries S i for i = 1;:::;n satisfying the Cuntz relations: S i S j = i;j 1 and n X i=1 S i S i = 1 The endomorphisms are determined by their actions on the Cuntz generators since they may be constructed so that the actions are preserved under the weak closure; specically, Izumi and Evans-Gannon obtain classication parameters by analyzing the possible images of the Cuntz generators under the irreducible sectors (i.e. the simple objects in the category). Suppose we have the following endomorphisms of a type III factor M giving sectors satisfying the near group fusion rules of type (G;m): 2 End(M) a self-conjugate, irreducible endomorphism G! End(M) an outer action of a nite group G on M, where we denote the action of g2G on v2M simply by g(v) Supposing further that H 2 (G;T) = 1 it is proven in [EvGan12, Theorem 1] that there are jGj +m isometries S g ;T 2M (with g2G and 2 where is an index set of order m) satisfying the Cuntz relations such that g = and g = Ad Ug where U : G! M is a unitary representation given by: U g = X h2G hg;hiS h S h + X 2 u g ( )T T g 40 whereh;i is a complex bicharacter on G, u g ( )2 C, and G acts on by permutation. They then provide formulae for the images of and the action of g2 G on the Cuntz algebra generators, and these give a complete invariant for the tensor equivalence classes corresponding to a given near-group fusion rule: Theorem A.2.1. [EvGan12, Theorem 1 & Corollary 1] Suppose that H 2 (G;T) = 1. Then the images of the Cuntz generators are as follows: g(S h ) =S g+h g(T ) = ~ (g)T for some ~ 2 b G (S g ) = s qdim() P h2G hg;hiS h + P ;2 u g ( )a(g ;)T T ! U g (T ) = P h2G;2 ~ (h)b ()S h T + P h2G;2 ~ (h)b 0 ()T S h S h + P ;;2 b 00 (;;)T T T where s2f1g a is a complex function on , and b ;b 0 ;b 00 are complex functions on , , and , respectively, for each 2 . Furthermore, the datafG;h;i;s;u g ;a;b ;b 0 ;b 00 g form a complete invariant for each tensor equivalence class of near groups of type (G;m). Using this result Evans-Gannon obtain a complete classication by identifying systems of equations that the classication datafh;i;s;u g ;a;b ;b 0 ;b 00 g for a given near-group fusion rule (G;m) must satisfy. Then solution sets to these systems are in correspondence with tensor equivalence classes of fusion categories having near group fusion rule (G;m). In fact, the 6j symbols may be obtained from the formulae for the images of the Cuntz generators in Theorem A.2.1, as was demonstrated for type (Z=(2); 2) near-groups by Suzuki and Wakui [SW, pages 60-63] in their computation of Turaev-Viro-Ocneanu invariants. However, this introduces more complicated systems of equations. The sector approach allows a classication to be obtained without the need to classify possible 6j symbols directly as was done in [TY]. 41 2.1 Type m =jGj1 near-groups LetC be a near group of type (G;jGj 1). By [EvGan12, Theorem 2(b)] we see that in this class FPdim() =jGj, the bicharacterh;i is identically 1, the action ofG on is trivial (g = 8g2G; 2 ), and we have the identication: ! b Gnf"g where " is the trivial character. By [EvGan12, Proposition 2] the only non-empty tensor equivalence classes of near groups of this type are those where G =F jGj+1 . Hence G is cyclic, and thus H 2 (G;T) = 1. Let p = char(F jGj+1 ) and let be the Kronecker delta function: x;y = 8 < : 1 ifx =y 0 ifx6=y By [EvGan12, Theorem 3(a)] near-groups of this type are in correspondence with solutions systems of equations involving the following parameters: a permutation of = b Gnf"g a 2f1;sg b : b Gnf"g!T b 00 : b Gnf"g b Gnf"g!T[f0g such that b 00 (;) = 0 () =" These give expressions for the complex parameters from Theorem A.2.1: u g ( ) =( )(g) a( ;) = 1 p jGj ; 1a b () = 1 p jGj ;( ) b( ) b 0 () =s ; 2 a b( 1 ) b 00 (;;) = ;()() ; b 00 (;) The tensor equivalence class for (G;jGj 1) given by the category of representations Rep(AGL 1 (F jGj+1 )) for the degree one ane general linear group over F jGj+1 corresponds to the solution s = a = b = b 00 = 1, and in fact, by [EGO, Corollary 7.4] and [EvGan12, Proposition 5], this is the only equivalence class of this type unlessjGj = 1; 2; 3; or 7. 42 2.2 Type m =jGj near-groups LetC be a near group of type (G;jGj). By [EvGan12, Theorem 2(c)] we have that the bicharacterh;i is non-degenerate and we may make the following identication: !G = b G The action ofG on is now given by left multiplication. The latter isomorphism G = b G is a consequence of G being abelian and is given by the bicharacterh;i (see Chapter 5). The group G now acts on the Cuntz generators T g via g(T h ) =hg;hiT h i.e., in the language of Theorem A.2.1, we let 7! g in the above correspondence via ~ (h) =hg;hi. The other parameters from Theorem A.2.1 are from [Iz00ii,x5, p. 623-624] and [EvGan12, Corollary 5] as follows: s = 1 u g (h) = 18g;h2G a(g;h) = g;h 1 p qdim() a(g) b g (h) = c p jGjqdim() hg;hi b 0 g (h) = 1 p jGj a(g)chg;hi b 00 g (h;k;l) = l;hk a(k)b(gk)hg;li for some c2C and complex functions a;b : =G!C satisfying the equations (5.1)-(5.4) in [Iz00ii,x5]. Moreover, again the set of solutions to these equations are in correspondence with the fusion categories of this type. 3 Center data from the Ocneanu tube algebra The tube algebra was introduced by Ocneanu for fusion categories of bimodules arising as standard invariants of a subfactor. Its denition was subsequently translated into the language of sectors by Izumi in [Iz00i, x3]; this is the formulation we give below. The important fact here is that, given a fusion category (of bimodules or sectors), the S and T 43 matrices for the Drinfel'd centerZ(C) may be obtained from the matrix units of the center of the tube algebra ofC. Denition A.3.1. LetC be a fusion category of sectors on a type III factor M. Then: Tube(C) := M ;;2Irr(C) Hom C (;) is a C -algebra [Iz00i, Proposition 3.2] where for x;y2Tube(C) the (;;)-component is denoted x[;;], multiplication is given by xy[;;] := 1 qdim() p qdim(C) X ;; 2Irr(C) qdim() qdim( ) N ; X i=1 0 B B B B B B B B B B B B B B @ p i ; y[;;] x[; ;] (p i ; ) 1 C C C C C C C C C C C C C C A wherefp i ; g N ; i=1 is a basis for Hom(; ), and the adjoint is given by: x[;;] := qdim() x 2 Hom( ; ) (Note that the multiplication is independent of the choice of basis.) Now consider the following subalgebra: Tube(C) 0 := M ;2Irr(C) Hom(;) and dene the following linear operator S 0 on it: S 0 (x[;;]) = qdim() x 2 Hom( ; ) 44 The center Z(Tube(C)) is evidently contained in Tube(C) 0 and by [Iz00i, Proposition 5.2] the operator S 0 leaves the center invariant. Next dene the following element t2Tube(C) given by: t[; ;] = qdim() By [Iz00i, Theorem 3.3(ii)] we have t 2 Z(Tube(C)). Now we may state the following theorem of Izumi: Theorem A.3.2. [Iz00i, Theorem 5.5] Dene the following two operators on Z(Tube(C)): S :=S 0 Z(Tube(C)) and T :=L t (left multiplication by t) Then the matrices forS andT are precisely the modular data for the Drinfel'd centerZ(C). By the previous section we know the form of the important intertwiner spaces for near- group fusion categories of sectors in terms of the Cuntz generatorsfS g ;T g g2G; 2 and the unitary representation U of the group G: Hom(gh;gh) =C1 Hom(;g) =C1 Hom(;g) =CU g Hom(g; 2 ) =CS g Hom(; 2 ) = M 2 CT Hence basic techniques from the theory of C algebras may be employed to compute the modular data for the centers of the fusion categories of interest in terms of the classication parameters of Evans-Gannon and Izumi. This is precisely how the modular data for centers of near groups were obtained in [EvGan12] and [Iz00ii]. 45 Bibliography [BaKi] B. Bakalov, A. Kirillov, Jr., Lectures on Tensor Categories and Modular Func- tors, University Lecture Notes 21 (2001) American Mathematical Society. [BasJ] T. Basak, R. Johnson, Indicators of Tambara-Yamagami categories and Gauss sums, Algebra Number Theory 9, no. 8 (2015) 1793-1823 [BKLR] M. Bischo, Y. Kawahigashi, R. Longo, K.-H. Rehren, Tensor Categories and Endomorphisms of von Neumann Algebras with Applications to Quantum Field Theory, Springer Briefs in Mathematical Physics 3, Springer, New York, NY (2015) [EG] P. Etingof, S. Gelaki, Isocategorical groups, Internat. Math. Res. Notices 2 (2001), 59-76. [EGNO] P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, Tensor Categories, Mathematical Surveys and Monographs 205, American Mathematical Society, Providence, RI (2015) [EGO] P. Etingof, S. Gelaki, V. Ostrik, Classication of fusion categories of dimension pq, Int. Math. Res. Notices 2004, no. 57 (2004) 3041{3056 [ENO] P. Etingof, D. Nikshych, V. Ostrik, On fusion categories, Ann. Math. 162 (2) (2005) 581{642. [EvGan12] D. Evans, T. Gannon, Near group fusion categories and their doubles, Adv. Math. 255 (2014), 586-640 [EvGan15] D. Evans, T. Gannon, Non-unitary fusion categories and their doubles via endomorphisms, arXiv:1506.03546 (2015) [EvKa] D. Evans, Y. Kawahigashi, Quantum symmetries on operator algebras, Oxford Mathematical Monographs, Oxford Science Publications/The Claren- don Press/Oxford University Press, New York, NY (1998) [I] I.M. Isaacs, Character Theory of Finite Groups, Dover Publications, New York, NY (1976) 46 [Iz91] M. Izumi, Application of Fusion Rules to Classication of Subfactors, Publi. RIMS, Kyoto Univ. 27 (1991) 953-994. [Iz00i] M. Izumi, The Structure of Sectors Associated with Longo-Rehren Inclusions I. General Theory, Commun. Math. Phys. 213 (2000), 127-179. [Iz00ii] M. Izumi, The Structure of Sectors Associated with Longo-Rehren Inclusions II. Examples, Rev. Math. Phys. 13, no. 603 (2001), 603{674 [Iz15] M. Izumi, A Cuntz algebra approach to the classication of near-group cate- gories, arXiv:1512.04288 [J] V.F.R. Jones, Index for subfactors. Invent. Math. 72, no. 1 (1983), 1-25. [Kas] C. Kassel, Quantum Groups, Graduate Texts in Mathematics 155, Springer- Verlag, New York, NY (1995) [LM] L. Linchenko, S. Montgomery, A Frobenius-Schur theorem for Hopf algebras, J. of Algebras and Representation Theory 3 (2000), 347{355. [Lo] R. Longo, Index of subfactors and statistics of quantum elds I., II., Commun. Math. Phys. 126 (1989) 217-247; 130 (1990) 285{309 [Mac] S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Graduate Texts in Mathematics 5, Springer-Verlag, New York, NY (1998) [MaN] G. Mason, S.-H. Ng, Central invariants and Frobenius-Schur indicators for semisimple quasi-Hopf algebras, Adv. in Math. 190 (2005), no. 1, 161-195 [Mas] A. Masuoka, Semisimple Hopf algebras of dimension 6, 8, Israel J. Math. 92 (1995), no. 1, 361{373 [M] S. 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Abstract (if available)
Abstract
Ng and Schauenburg generalized higher Frobenius-Schur indicators to pivotal fusion categories and showed that these indicators may be computed utilizing the modular data of the Drinfel'd center of the given category. We consider two classes of fusion categories generated by a single non-invertible simple object: near-groups, those fusion categories with one non-invertible object, and Haagerup-Izumi categories, those with one non-invertible object for every invertible object. Examples of both types arise as representations of finite or quantum groups or as Jones standard invariants of finite-depth Murray-von Neumann subfactors. We utilize the Evans-Gannon computation of the tube algebras to obtain formulae for the Frobenius-Schur indicators of objects in both of these families.
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Frobenius-Schur indicators for near-group and Haagerup-Izumi fusion categories
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