On Lagrangian Algebras in Braided Fusion Categories

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A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University

In partial fulﬁllment of the requirements for the degree Doctor of Philosophy

Darren Allen Simmons April 2017

This dissertation titled On Lagrangian Algebras in Braided Fusion Categories

by DARREN ALLEN SIMMONS

has been approved for the Department of Mathematics and the College of Arts and Sciences by

Alexei Davydov Assistant Professor of Mathematics

Robert Frank Dean, College of Arts and Sciences 3 Abstract

SIMMONS, DARREN ALLEN, Ph.D., April 2017, Pure Mathematics On Lagrangian Algebras in Braided Fusion Categories (147 pp.) Director of Dissertation: Alexei Davydov This work is part of an ongoing study of a generalization of the notion of symmetry. In mathematics, the notion of symmetry is formalized into the concept of a group. Examples of groups arise naturally throughout mathematics, chemistry, physics, and other ﬁelds. However, recent developments in high-energy and condensed matter physics show that not all kinds of symmetry that arise in nature can be captured by groups. A versatile, more general tool that does this job is a certain class of tensor categories known as fusion categories. We add to the study of these categories by constructing certain analogues to the classical algebraic notion of an associative algebra over a ﬁeld inside them, and then expounding the properties of these generalized algebras. 4

For my parents, Al and Jennifer, without whose unwavering love and support none of this would have been possible. 5 Table of Contents

Page

Abstract...... 3

Dedication...... 4

List of Tables...... 8

0 Introduction...... 9 0.1 General Background and Motivation...... 9 0.2 Main Results...... 15 0.2.1 Executive Summary...... 15 0.2.2 More Details — Main Results of Chapter3...... 17 0.2.3 More Details — Main Results of Chapter4...... 18

1 Deﬁnitions...... 23 1.1 Categories, Part I, or “An Introduction to Abstract Nonsense”...... 23 1.1.1 Categories and Morphisms...... 23 1.1.2 Functors...... 25 1.1.3 Monoidal Categories and Braiding...... 25 1.1.4 Invertible Objects and Pointed Categories...... 26 1.1.5 Rigid, Fusion, and Ribbon Categories...... 27 1.1.6 Pivotal and Spherical Categories...... 27 1.2 All About Algebras, or “So You Want to Multiply”...... 29 1.2.1 Algebras and Modules...... 29 1.2.2 The Full Center of an Algebra...... 31 1.3 Group Cohomology...... 32 1.3.1 G-modules and the Standard Normalized Complex...... 32 1.3.2 Abelian Cohomology...... 34 1.4 Categories, Part II, or “Generalized Abstract Nonsense”...... 35 1.4.1 Graded Vector Spaces...... 35 1.4.2 The Category V(G, α)...... 36 1.4.3 The Category Z(G, α)...... 37 1.4.4 The Category C(G, H, α)...... 39 1.4.5 The Category C(A, q) and Quadratic Functions...... 39 1.4.6 Categorical Groups and Their Standard Invariants...... 40 1.4.7 Module Categories...... 41 1.4.8 Module Functors...... 41 1.4.9 Module Natural Transformations...... 42 1.4.10 Simple V(G, α)-module Categories...... 42 6

2 Preliminary Results...... 44 2.1 Algebras in Monoidal Categories...... 44 2.1.1 Maschke’s Lemma...... 44 2.1.2 The Simple Things...... 46 2.1.3 The Full Center of an Algebra...... 46 2.2 Commutative Algebras in Group-theoretical Categories...... 48 2.2.1 Group-theoretical Braided Fusion Categories...... 48 2.2.2 Ribbon Structure on Group-theoretical Braided Fusion Categories. 48 2.2.3 The Monoidal Center of V(G, α)...... 50

3 Lagrangian Algebras in Group-theoretical Braided Fusion Categories...... 53 3.1 Description of the Category Z(G, α)...... 53 3.1.1 Simple Objects of Z(G, α)...... 53 3.2 Algebras in Group-theoretical Modular Categories...... 58 3.2.1 Algebras in the Category Z(G, α)...... 58 3.2.2 Commutative Separable Algebras in Trivial Degree and Their Modules...... 59 3.2.3 Commutative Separable Algebras Trivial in Trivial Degree and Their Local Modules...... 68 3.2.4 Etale Algebras and Their Local Modules...... 76 3.3 The Full Center of an Algebra Redux...... 77 3.3.1 The Full Center of the Twisted Group Algebra...... 77 3.4 Characters of Lagrangian Algebras...... 79 3.4.1 Irreducible Characters...... 79 3.4.2 Characters of Lagrangian Algebras in Z(G, α)...... 82 3.5 Example: Characters of Z(D3, α) ...... 86 3 ∗ 3.5.1 Brief Description of H (D3, C ) ...... 86 3.5.2 Wherein |α| = 1...... 86 3.5.3 Wherein |α| = 2...... 88 3.5.4 Wherein |α| = 3...... 89 3.5.5 Wherein |α| = 6...... 90 3.6 Automorphisms of Lagrangian Algebras in Group-theoretical Braided Fusion Categories...... 92 3.6.1 Lagrangian Algebras in Drinfeld Centers of Finite Groups..... 92 3.6.2 Automorphisms of Lagrangian Algebras...... 92

4 Lagrangian Algebras in Pointed Categories...... 95 4.1 Modular Extensions of Finite Groups...... 95 4.1.1 The Categorical Group Mex(S)...... 95 4.1.2 Invertible Objects of Z(G, α)...... 100 4.2 Lagrangian Extensions of Abelian Groups...... 103 4.2.1 The Categorical Group Lex(B)...... 103 7

4.3 The Categorical Group Ext(B, D)...... 112 4.3.1 Wherein the Arguments of Ext are Dual to One Another...... 114 4.4 Comparing Lex(B) with Ext(bB, B)...... 119 4.4.1 F ’n’ φ ...... 119 4.5 Explicit Computation of the Homomorphism Lex(L) → H3(bL, k∗)..... 125 4.5.1 Lagrangian Algebras in Pointed Categories...... 126 4.6 Examples...... 136

Appendix: Homotopy Action on the Standard Normalized Complex...... 143 8 List of Tables

Table Page

3.1 Characters of Z(D3) ...... 87 3.2 Characters of Lagrangian Algebras in Z(D3) ...... 87 3.3 Decomposition into Irreducible Characters...... 88 3.4 Characters of Z(D3, α); |α| = 2...... 88 3.5 Characters of Lagrangian Algebras in Z(D3, α) ...... 89 3.6 Decomposition into Irreducible Characters...... 89 3.7 Characters of Z(D3, α); |α| = 3...... 89 3.8 Characters of Lagrangian Algebras in Z(D3, α) ...... 90 3.9 Decomposition into Irreducible Characters...... 90 3.10 Characters of Z(D3, α); |α| = 6...... 91 3.11 Characters of Lagrangian Algebras in Z(D3, α) ...... 91 3.12 Decomposition into Irreducible Characters...... 91 9 0 Introduction

0.1 General Background and Motivation

The concept of a group underlies the entirety of abstract algebra. Virtually all of the algebraic structures that are studied—groups (obviously), rings, modules, fields, vector spaces, et cetera—can be thought of as groups with some additional structure imposed thereon. Consequently, the theory of groups is the foundation upon which is built all of modern algebra, so by studying groups, we study algebraic structures in general. Groups arise naturally throughout mathematics, chemistry1, physics, and other fields of study as abstract generalizations of the notion of symmetry. Some are quite concrete—e.g., the group of geometric symmetries of a polygon or polyhedron—while others are far more abstract—e.g., the Galois group of admissible permutations of the roots of a polynomial. That latter example, of course, dates back to Galois and was one of the crucial factors behind the historical development of the theory of groups. Galois himself was only interested in understanding why the general polynomial equations of degree 5 or higher were not solvable by radicals, but the theory that he developed2 to answer this question was the first step towards the abstract study of groups for their own sake. The primary motivation behind this work is the study of “generalized groups” in some sense. Specifically, we study braided fusion categories, which are the “generalized groups” to which we referred. One might ask, “Why study categories if your real interest is groups?” One (admittedly complicated) answer is as follows: there is a limitation to the utility of groups as models of symmetry as it exists in the real world, at least at the quantum level. As it was noted in [30], certain topological states of matter3 are classified by categories of a certain type. In particular, particle-like excitations correspond to objects

1 E.g., the VSEPR theory of molecular bonding. 2 Which now bears his name. 3 These are called “2+1D topological orders”. See [30]. 10 of a nondegenerate unitary braided fusion categories, with the vacuum corresponding to the monoidal unit. This gives one possible answer to the natural question of “Why categories?”—just as groups arise naturally in the study of symmetry in the real world, so braided fusion categories arise in the study of symmetry in the quantum world. In modern mathematics, we often study groups via their actions on other objects. This is the idea underlying classical representation theory, wherein a group is studied by allowing it to act on a vector space by linear transformations. Just as the class of all finite-dimensional vector spaces over a given field k, together with linear transformations, forms a category, so too does the collection of all finite-dimensional k-linear representations of a given finite group G. This category, which we call Rep(G), can be interpreted as the category of finite-dimensional vector spaces with k-linear G-action, with G-invariant linear transformations as its morphisms. In this interpretation, Rep(G) is naturally a subcategory of k-Vect. A rich theory has been developed that describes the category Rep(G). E.g., character theory arises from the notion of the trace of a linear operator from linear algebra as applied to the linear action of a group on a finite-dimensional vector space. The category of finite-dimensional vector spaces is naturally endowed with a “formal addition”: the direct sum. On the level of dimension, the direct sum is indeed additive. The direct sum also has an identity object: the trivial vector space {0}. Thus the collection of (isomorphism classes of) vector spaces forms a monoid with respect to the direct sum. Moreover, every vector space is a direct sum of finitely many copies of the simple vector space k. Thus k-Vect is a semisimple category—an Abelian category for which all objects are direct sums of simple objects. It follows from Maschke’s lemma4 that the category Rep(G) is semisimple precisely when the characteristic of k does not divide the order of G.

4 Lemma 2.1.1.1( q.v.). 11

k-Vect is also endowed with a “formal multiplication”: the tensor product, which is also multiplicative on the level of dimension. The tensor product also has an identity: the ﬁeld k itself, i.e., the isomorphism class of the one-dimensional space. Since the tensor product is also associative, the tensor product of vector spaces makes k-Vect into the simplest example of what we call a monoidal category. This is a category equipped with a tensor product that has a unit object and is associative up to a natural isomorphism called the associator. For k-Vect, the associator is trivial, since the tensor product of vector spaces is associative on the nose. The diagonal action allows us to make the tensor product of representations into a representation in its own right. This turns Rep(G) into a monoidal category, with the monoidal unit being the trivial representation. The tensor product in Rep(G) is also associative on the nose, and so the associator of Rep(G) is also trivial. The category of ﬁnite-dimensional vector spaces also has dual objects. To each vector space V, there is associated the vector space V∗ of linear functionals on V. The dual vector space comes equipped with two natural maps: the evaluation map—which takes a simple tensor `⊗v ∈ V∗⊗V and evaluates5 the functional ` at the vector v—and the coevaluation map—which takes a scalar λ ∈ k and returns the sum

dimk(V) X i λei⊗e , i=1 n j o where {ei|i = 1, 2,..., dimk(V)} is any basis for V and e j = 1, 2,..., dimk(V) is the corresponding dual basis6. The existence of these dual objects makes k-Vect a rigid category. Colloquially, a monoidal category is called fusion if it has (left) duals, every object is a direct sum of simple objects, and the collection of (isomorphism classes of) simple objects is ﬁnite. Fusion categories are generalized groups in the sense that the collection

5 Hence the name. 6 Surprisingly enough, the value of the coevaluation map does not depend upon the choice of basis. 12 of simple objects of a fusion category forms a group under the tensor product. The simplest possible example of a fusion category is not Rep(G), but rather something else: the category of G-graded vector spaces. The collection of G-graded vector spaces, along with linear maps that preserve the grading, forms a category that we denote by V(G). The simple objects of V(G) are one-dimensional vector spaces concentrated in a single degree. That degree is, in turn, labeled by a single group element. Consequently, there is a one-to-one correspondence between simple objects of V(G) and elements of G. In particular, when G is finite, so too is the set of (isomorphism classes of) simple objects of V(G). The tensor product of vector spaces extends7 to G-graded vector spaces in a natural way in terms of the group operation on G. The unit object is a one-dimensional vector space concentrated in the degree labeled by the identity element e ∈ G. Applied to simple objects I(g) and I(h) of V(G), where g, h ∈ G, the tensor product I(g)⊗I(h) is isomorphic to I(gh). Moreover, the notion of the dual of a vector space passes8 without change to the category of G-graded vector spaces. It follows from this discussion that the category of G-graded vector spaces is a fusion category. Indeed, on the level of simple objects, V(G) is just G. In other words, V(G) is a proper-class-sized model of a finite object. V(G) also has a rather unusual property: despite the fact that V(G) is a subcategory of k-Vect, the dimension of V(G) is |G|, while the dimension of k-Vect is 1. The theory of fusion categories has applications in representation theory, theoretical physics, and quantum computing. Consequently, it is, as of the time of this writing, experiencing a period of rapid development. Our interest in fusion categories stems from the fact that the class of fusion categories forms a nice generalization of the notion of groups. They almost—but not quite—fit the mold for this generalized symmetry that we

7 See § 1.4.1. 8 See § 2.2.3. 13 wish to describe. We are interested in a special class of fusion categories that come equipped with an extra structure—a braiding. This additional structure can be seen as an answer to a natural question about the tensor product in a monoidal category. That question is, “Given two objects X and Y, is X⊗Y isomorphic to Y⊗X?” Put more simply, the question amounts to “Is the tensor product commutative or not?” The answer, in general, is that it can be, but it does not have to be. Moreover, there are two possible interpretations of the meaning of the word “commutative” in the context of a tensor category. One is called symmetry, and the other, nondegeneracy. Colloquially, a symmetry is a swap that, when done twice, is akin to doing nothing. Also colloquially, a swap is nondegenerate if it is as far from being symmetric as it can possibly be. The former is what happens in the category of vector spaces9. The latter is what we want the objects of our study to satisfy. We call such swaps braidings. The terms “symmetry” and “braiding” come from the notions of the symmetric and braid groups, respectively, and the actions of these groups on tensor factors in a monoidal category. In short, symmetries are symmetric because the square of a generator of the symmetric group S n is the identity, whereas braidings are nondegenerate because the square of a generator of the braid group Bn is not the identity. One method of producing examples of nondegenerate categories is by the so-called monoidal center construction—the categoriﬁcation of a classical algebraic construction. This is an analogue, at the level of categories, of the notion of the center of a monoid, group, or ring. Given a monoidal category C, we look at the collection of all those objects Z ∈ C for which Z⊗X is isomorphic to X⊗Z for all X ∈ C. This collection forms a subcategory Z(C) of C called the monoidal center. The monoidal center Z(C) has the property that, for any two objects X, Y ∈ Z(C), X⊗Y is isomorphic to Y⊗X. Z(C) is our archetype for a braided monoidal category: a monoidal category with a braiding.

9 This is because isomorphism classes of vector spaces correspond to cardinal numbers, multiplication of cardinal numbers is commutative, and the tensor product is multiplicative on the level of dimension. 14

When we apply the monoidal center construction to the category of G-graded vector spaces, the result is a braided fusion category which we call the Drinfeld center of G. The Drinfeld center Z(G) also has a rather odd property that is at odds with what usually happens in algebra when we compute the center of something. Namely, the dimension of the Drinfeld center Z(G) is |G|2, while the dimension of V(G) is |G|. It is for this reason that the Drinfeld center is sometimes called the Drinfeld double. Unusually among algebraic objects, the category V(G) gets larger (in some sense) when you compute its monoidal center, rather than smaller. Another nice consequence is that the braiding10 in the Drinfeld center Z(G) is always nondegenerate (and hence not symmetric) so long as G is not the trivial group. The braiding in Z(G) amounts to the structure of a G-action on the objects of Z(G). Thus we can realize11 the category Z(G) as the category whose objects are G-graded vector spaces with compatible G-action, and whose morphisms are linear maps that preserve both the grading and the G-action. We use this interpretation as our deﬁnition of Z(G), and prove later12 that Z(G) is equivalent to the monoidal center of the category V(G) of graded vector spaces. The tensor product of vector spaces is naturally associative on the nose, but for graded vector spaces, it need not be. Instead, a sort of “skew associativity” can be imposed via a function α : G × G × G → k∗ that appears as a constraint: h i u⊗(v⊗w) = α( f, g, h) (u⊗v)⊗w , ∀u ∈ U f , ∀v ∈ Vg, ∀w ∈ Wh for G-graded vector spaces U, V, and W. It can be shown that such a function α must satisfy the 3-cocycle13 condition

α(y, z, w)α(x, yz, w)α(x, y, z) = α(xy, z, w)α(x, y, zw), ∀x, y, z, w ∈ G.

10 See § 1.4.3 for the deﬁnition of the braiding. 11 See § 1.4.3. 12 In chapter2, proposition 2.2.3.3. 13 See § 1.3.1. 15

If we apply the monoidal center construction to the category V(G, α) of G-graded vector spaces with skew associativity controlled by α, then we obtain the twisted Drinfeld center Z(G, α), whose objects are G-graded vector spaces with projective G-action, and whose morphisms are k-linear maps preserving the grading and projective action. The notion of an associative unital algebra also passes14 from the category k-Vect to arbitrary monoidal categories C. In particular, this notion passes to the categories Rep(G), V(G, α), and Z(G, α). The twisted associativity in the latter two categories makes these algebras skew associative in an absolute sense, but associative within their indicated categories15. The braiding in a braided category C also allows us to consider commutative algebras in C—in particular, in the braided category Z(G, α). As with algebras in the classical sense, we can also consider the categories of modules over these algebras, as well as local modules. Local modules16 are those modules over a commutative algebra in a braided category that are unaﬀected by a double braiding swap17 prior to multiplication. There is no classical algebraic analogue for this notion. Rather, it is a purely quantum18 phenomenon. Those commutative algebras whose categories of local modules are trivial in some sense19 we call Lagrangian algebras.

0.2 Main Results

0.2.1 Executive Summary

In this work, we study Lagrangian algebras in braided fusion categories. One reason for which we choose to study these algebras in this environment is that Lagrangian

14 See § 1.2.1. 15 See §§ 1.2.1 and 1.4.3. 16 This is unrelated to the usual algebraic notion of a local module. 17 See the diagram (1.1) in § 1.2.1. 18 This is explained to some degree in the introduction of [30]. 19 Namely, in the sense that they are equivalent to k-Vect. See § 1.2.1. 16 algebras are (labels for) modular invariants for rational conformal field theories20. Another reason is that they allow us to identify a given nondegenerate braided fusion category with the monoidal center of a finite group. In this sense, Lagrangian algebras can be thought of as being analogous to a choice of coordinates in linear algebra. In chapters1 and2, we build up the necessary background to discuss the main objects of our study. In chapter3, we give a classification of Lagrangian algebras in group-theoretical braided fusion categories: Lagrangian algebras in the Drinfeld center Z(G, α) correspond to pairs (H, γ), where H ≤ G is a subgroup, and γ ∈ C2(G, k∗) is a coboundary for the restriction of α to H. The aforementioned classification is the main result of chapter3. We then use the classification to compute the characters of Lagrangian algebras in general, and for a tractable concrete example. It is known that Lagrangian algebras in pointed braided fusion categories corresponded to Lagrangian subgroups. It is also not difficult to compute the automorphism groups of these Lagrangian algebras. Another quantum phenomenon equips these Lagrangian algebras with associators—classes in the third cohomology of the grading group of the category21. However, there was no known formula for the associator of the group of symmetries of such a Lagrangian algebra. The main result of chapter4 is the formula for the associator that we derive in § 4.5. As a byproduct of describing the associator, we also obtain a description of the third cohomology group of an arbitrary

ﬁnite Abelian group B in terms of Lagrangian extensions of the character group bB, the group of antisymmetric 3-linear forms on B, the 2-torsion subgroup B2 of B, Abelian group extensions of B by bB, and the quotient group B/2B.

20 See, e.g.,[5]. 21 See § 1.4.5. 17

0.2.2 More Details — Main Results of Chapter3

The classiﬁcation of Lagrangian algebras from chapter3, in particular, gives a method for explicitly computing physical modular invariants22 for group-theoretical modular data. It should be mentioned that the set of labels for physical modular invariants was obtained in [39] using the language of module categories. By establishing a correspondence between Lagrangian algebras and module categories, and by computing the characters of Lagrangian algebras, we give a method for determining modular invariants corresponding to module categories. Indeed, computing modular invariants becomes nothing more or less than knowing Lagrangian algebras in a group-theoretical braided fusion category. We classify Lagrangian and the more general etale algebras in the Drinfeld center Z(G, α) in two steps. First, we describe etale algebras with the trivial grading. These are nothing but indecomposable commutative separable algebras with a G-action, and hence23 they are just function algebras on transitive G-sets. Up to isomorphism, they are labeled by conjugacy classes of subgroups H ≤ G. Then we identify the category of local modules

loc Z(G, α)k(G/H) with the group-theoretical modular category Z(H, α|H). A general etale algebra in Z(G, α) is an extension of its trivial degree component, and hence is an etale

loc algebra in one of Z(G, α)k(G/H). Considered as an algebra in Z(H, α|H), its trivial-degree component is one-dimensional. Our second step is to classify such algebras and their categories of local modules. Then we combine the results obtaining the description of all etale algebras in Z(G, α) and their local modules. As a corollary, we get a classiﬁcation of Lagrangian algebras in Z(G, α): they are parametrized by conjugacy classes of subgroups

H ≤ G together with a coboundary ∂γ = α|H, matching with the answer from [39]. We explain this matching by identifying Lagrangian algebras with full centers of

22 This problem was raised in [5]. 23 We work over an algebraically closed ﬁeld k. See lemma 3.2.2.1. 18 indecomposable separable algebras in V(G, α). Finally, after recalling the character theory for objects of Z(G, α), we compute characters of Lagrangian algebras. We treat the

case G = D3—the dihedral group of order 6—as an example.

0.2.3 More Details — Main Results of Chapter4

The third Abelian cohomology classifies24 braided pointed fusion categories. Such categories are labeled by triples (A, α, c), where A is an Abelian group, α is a 3-cocycle of A with coefficients in k∗, and c is a 2-cochain of A with coefficients in k∗ such that the pair

25 3 ∗ (α, c) belongs to the third Abelian cohomology group Hab(A, k ). Moreover, the Eilenberg–Mac Lane interpretation of the third Abelian cohomology26 as the group of quadratic forms has a natural manifestation on level of categories: braided pointed fusion categories can instead be labeled by pairs (A, q), where A is as above and q : A → k∗ is the (unique) quadratic function corresponding to the Abelian 3-cocycle (α, c). Associators for a pointed fusion category are classiﬁed by the third cohomology group H3(G, k∗). These are associativity constraints for the tensor product of G-graded vector spaces. Recently, the group structure of the third cohomology was interpreted27 as the group of modular extensions of the category Rep(G) of ﬁnite-dimensional k-linear representations of G. Lagrangian algebras in modular categories seem to play a very special role. For example, a Lagrangian algebra L ∈ C allows us to identify C with the monoidal center

Z(CL) of the category CL of L-modules in C. The group of invertible modules over a Lagrangian algebra comes equipped with a natural 3-cocycle (the associator). The group of invertible modules over a Lagrangian algebra contains a great deal of information about

24 See [26, Proposition 3.1]. 25 See § 1.3.2. More details can also be found in [17] and [26]. 26 See [17]. 27 In [30]. 19 the ambient category. For example, when all simple modules over a Lagrangian algebra are invertible, the ambient category can be identiﬁed with a twisted Drinfeld center28. In a modular pointed category, all Lagrangian algebras correspond to Lagrangian subgroups of the grading group. Moreover, all simple modules over Lagrangian algebras are invertible and the group of invertible modules is the quotient by the corresponding

29 Lagrangian subgroup. Thus a Lagrangian subgroup L A gives a braided equivalence

C(A, q) 'Z(A/L, β) for some β ∈ H3(A/L, k∗). The main purpose of chapter4 is to describe the associator β ∈ H3(A/L, k∗) as a function of A, q and L. We derive a formula for a cocycle representing it30, but the explicit expression is not very easy to use. E.g., the formula does not reveal the cohomology class of the associator. In order to find a more useful description of the associator, we look at the correspondence between triples (A, q, L) (a nondegenerate quadratic group and a Lagrangian subgroup) and the cohomology classes from H3(A/L, k∗) from a different perspective. By fixing L and varying (A, q), we turn this correspondence into a homomorphism of groups. We define a natural group structure on the set of isomorphism classes of Lagrangian extensions of L. Assigning the associator to the triple (A, q, L) becomes a homomorphism from the group Lex(L) of Lagrangian extensions to the third cohomology:

Lex(L) −→ H3(bL, k∗) . (†)

Here, we use the natural identiﬁcation of A/L with the character group bL induced by the quadratic function on A. The homomorphism (†) is a compact way of expressing the associator β ∈ H3(A/L, k∗) as a function of A, q and L.

28 See chapter3 or [12] for more. 29 Here, we abuse notation by referring to the Lagrangian subgroup as L also. 30 See § 4.5. 20

Furthermore, the homomorphism (†) has a nice interpretation in terms of low-dimensional cohomology. It can be used to unpack the internal structure of the third cohomology of an Abelian group. To simplify the exposition, write bL = B. The homomorphism (†) is an embedding whose image is the kernel of the alternation map

3 ∗ 3 ∗ n Alt3 : H (B, k ) → HomZ Λ B, k . Here we denote by HomZ(Λ B, M) the group of antisymmetric n-linear forms on B with values in an Abelian group M. Our ground ﬁeld k is algebraically closed of characteristic zero; consequently, its multiplicative group k∗ is a

∗ divisible group. The torsion subgroup of k is isomorphic to Q/Z. Thus for ﬁnite B, we ∗ can trade k for the universal torsion group Q/Z as coeﬃcient groups for cohomology without changing the isomorphism classes of the cohomology groups.

n The functor B 7→ H (B, Q/Z) is polynomial of degree n. As a polynomial functor, it has a ﬁltration whose associated graded components are homogeneous polynomial functors. For degrees one and two, the cohomology is already homogeneous:

1 2 2 H (B, Q/Z) = bB, H (B, Q/Z) = HomZ Λ B, Q/Z .

The degree-three cohomology is a combination of homogeneous polynomial functors of degrees three and two. When B is 2-torsion-free, the third cohomology group ﬁts in the split exact sequence

Alt3 {0} / Ext1 (B, B)τ / H3(B, / ) / Hom Λ3B, / / {0} , Z b Q Z Z Q Z

1 B, B τ → H3 B, / with the map ExtZ( b) ( Q Z) given by factoring through the embedding 1 B, B ,→ H2 B, B γ 7→ γ x, y z x, y, z ∈ B ExtZ( b) ( b) and then assigning ( )( ) for . 21

For a general ﬁnite Abelian group B, the situation is a bit more complicated. We have a diagram with exact rows and columns:

{0} (‡)

Bb2

ψ 3 Alt3 3 {0} / Lex(bB) / H (B, Q/Z) / HomZ Λ B, Q/Z / {0}

φ Ext1 (B, B)τ Z b

B[/2B

{0}

In order to build the vertical sequence in the diagram (‡), we interpret both the collection of Lagrangian extensions of bB and the collection of extensions of B by bB as the categorical groups31 Lex(bB) and Ext(B, bB), respectively. The zeroth homotopy groups of Lex B Ext B, B B 1 B, B (b) and ( b) are Lex(b) and ExtZ( b), respectively. We construct a functor Lex(bB) → Ext(B, bB). This functor induces the indicated homomorphism φ B → 1 B, B B : Lex(b) ExtZ( b) from the diagram (‡). Note also that in the case where is B 1 B, B τ 2-torsion-free, all terms of the vertical sequence in (‡) except Lex(b) and ExtZ( b) vanish32.

The map ψ : Bb2 → Lex(bB) sends χ into a Lagrangian extension (B × bB, qstd + q˜, ι), whereq ˜ is a quadratic function B → Q/Z such that 2˜q(x) = χ(x) for all x ∈ B2. The map φ B → 1 B, B τ A, q, ι : Lex(b) ExtZ(b ) assigns to a Lagrangian extension ( ) an extension of 31 See, e.g.,[26, § 3.1]. 32 In that case, exactness of the vertical sequence gives an isomorphism Lex(bB) ' Ext1 (B, bB)τ. Z 22

ι π Abelian groups B −→ A −→ bB, where the projection π sends a ∈ A into (π(a))(x) = q(a + ι(x)) − q(a) for x ∈ B. See § 4.4 for more. 23 1 Definitions

Here, we review a number of preliminary concepts.

1.1 Categories, Part I, or “An Introduction to Abstract Nonsense”

1.1.1 Categories and Morphisms

A category C is a pair of classes C = Obj(C), Map(C) . Elements of Obj(C) are called objects and elements of Map(C) are called morphisms or arrows. We will follow the standard abuse of notation and write A ∈ C to denote that A is an object of the category C. For each morphism f ∈ Map(C), there exist uniquely determined33 A, B ∈ C, called the source object and target object, respectively. We write

f : A → B

and say that f is a morphism from A to B. We denote by HomC(A, B) the class of morphisms with source object A and target object B. For any three objects A, B, C ∈ C, there is a binary operation

◦ : HomC(B, C) × HomC(A, B) → HomC(A, C) called composition of morphisms. The composition of f : A → B and g : B → C is denoted by g ◦ f . For any category C, the following axioms hold:

• (Associativity): If f : A → B, g : B → C, and h : C → D are morphisms, then h ◦ (g ◦ f ) = (h ◦ g) ◦ f .

• (Identity): For every X ∈ C, there exists a morphism IdX : X → X such that, for any

morphisms f : A → X, g : X → B, we have IdX ◦ f = f and g ◦ IdX = g, respectively.

33 By f . 24

The morphism IdX is called the identity morphism. For any object X, it can be shown that the identity morphism IdX is unique. A morphism f : X → Y is a monomorphism if it is left cancellative, i.e., for all morphisms g1, g2 : Z → X, we have

( f ◦ g1 = f ◦ g2) =⇒ g1 = g2.

A monomorphism is normal if it is the kernel of some morphism. Dually, f : X → Y is an epimorphism if it is right cancellative, i.e., for all morphisms g1, g2 : Y → Z, we have

(g1 ◦ f = g2 ◦ f ) =⇒ g1 = g2.

An epimorphism is conormal if it is the cokernel of some morphism. If C is a category, then the opposite category to C, denoted by Cop, is the category whose objects are the objects of C, and whose morphisms are the morphisms of C with all arrows reversed. An object I ∈ C is an initial object if, for every object Y ∈ C, there is a unique morphism f : I → Y. Dually, an object T ∈ C is a terminal object if, for every object X ∈ C, there is a unique morphism g : X → T. An object 0 ∈ C that is both initial and terminal is called a zero object. A category C is Abelian if it has a zero object, it has all pullbacks and pushouts, all of its monomorphisms are normal, and all of its epimorphisms are conormal. An object X of an Abelian category C is simple if X is not isomorphic to the zero object 0 ∈ C and the only subobjects of X are 0 and X. The set of isomorphism classes of simple objects of C is denoted by Irr(C). An Abelian category C is semisimple if every object X ∈ C is a direct sum of simple objects. 25

1.1.2 Functors

Let C and D be categories. A functor F : C → D is a mapping such that each of the following hold:

• For each X ∈ C, F(X) ∈ D.

• For each f ∈ HomC(X, Y), F( f ) ∈ HomD(F(X), F(Y)), and

– F(IdX) = IdF(X) for all X ∈ C.

– F(g ◦ f ) = F(g) ◦ F( f ) for all morphisms f : X → Y, g : Y → Z.

1.1.3 Monoidal Categories and Braiding

A category C is called monoidal or tensor if there is a bifunctor

⊗ : Obj (C) × Obj (C) → Obj (C)

∼ that is associative up to a natural isomorphism α : A ⊗ (B ⊗ C) −→ (A ⊗ B) ⊗ C, called the associator, and for which there is an object I ∈ C, called the monoidal unit, such that, for ∼ ∼ all A ∈ C, there are isomorphisms λA : I⊗A −→ A and ρA : A⊗I −→ A, called the left and right unit maps, respectively. The monoidal unit I can be shown to be unique up to natural isomorphism. To say that ⊗ is a bifunctor means that, for any A, B ∈ C, each of A ⊗ − : C → C and − ⊗ B : C → C are functors. A monoidal category C is strict if each of α, λ, and ρ are identities. A monoidal category C is called braided if, for each pair of objects A, B ∈ C, there is an isomorphism

cA,B : A ⊗ B → B ⊗ A,

natural in A and B, that satisﬁes the hexagon axioms. By “satisﬁes the hexagon axioms”, we mean that the collection of braiding maps c(·),(·) makes both of the following diagrams 26 commute:

cA⊗B,C (A ⊗ B) ⊗ C / C ⊗ (A ⊗ B) 6 α α ( A ⊗ (B ⊗ C) (C ⊗ A) ⊗ B 6

Id ⊗c c ⊗Id A B,C ( A,C B A ⊗ (C ⊗ B) α / (A ⊗ C) ⊗ B

cA,B⊗C A ⊗ (B ⊗ C) / (B ⊗ C) ⊗ A 6 α−1 α−1 ( (A ⊗ B) ⊗ C B ⊗ (C ⊗ A) 6

c ⊗Id Id ⊗c A,B C ( B A,C (B ⊗ A) ⊗ C / B ⊗ (A ⊗ C) α−1

Let hC, ⊗, Ii be a monoidal category. The collection

Z(C) = Z ∈ C A⊗Z ' Z⊗A ∀A ∈ C forms a full braided subcategory of C that we call the monoidal center or Drinfeld center of C. For Z ∈ Z(C) and A ∈ C, there is a natural isomorphism zA : A⊗Z → Z⊗A that we call the half-braiding.

1.1.4 Invertible Objects and Pointed Categories

An object X of a monoidal category hC, ⊗, Ii is called invertible if there is an object ∼ ∼ Y ∈ C and isomorphisms X⊗Y −→ I, Y⊗X −→ I. The set of isomorphism classes of invertible objects of a monoidal category C forms a group under ⊗ that is called the Picard group of C and is denoted by Pic(C). A monoidal category C is called pointed if every simple object of C is invertible. The Picard subcategory Pic(C) of a monoidal category C is the full subcategory of C generated by all invertible objects. 27

1.1.5 Rigid, Fusion, and Ribbon Categories

A monoidal category hC, ⊗, Ii is rigid if, for every object X ∈ C, there is an object

∗ ∗ X ∈ C, called the (left) dual of X, and morphisms evX : X ⊗ X → I (the evaluation map)

∗ and coevX : I → X ⊗ X (the coevaluation map) such that each of the following compositions is the identity:

coev⊗Id α−1 Id⊗ev X / (X ⊗ X∗) ⊗ X / X ⊗ (X∗ ⊗ X) / X

Id⊗coev α ev⊗Id X∗ / X∗ ⊗ (X ⊗ X∗) / (X∗ ⊗ X) ⊗ X∗ / X∗

A k-linear monoidal category C is fusion if C is semisimple, C is rigid, and the collection Irr(C) of isomorphism classes of simple objects of C is ﬁnite. A rigid monoidal category C is a ribbon category if, for each object X ∈ C, there is a morphism θX : X → X, called the ribbon twist, such that θX⊗Y = cY,X ◦ cX,Y (θX⊗θY ).

1.1.6 Pivotal and Spherical Categories

Following [2], we say that a k-linear monoidal category C is a category with dual

∗ op ∗ ∗ data if there is a functor (·) : C → C, natural transformations τ : IdC → (·) ◦ (·) and γ : ((·)∗ × (·)∗) ◦ ⊗ → ⊗op ◦ (·)∗, and an isomorphism ν : I → I∗. Each component of the natural transformations τ and γ is required to be an isomorphism. If the dual data are coherent, then the category C is called a category with duals. A category with duals is

∗ called a pivotal category if, for each object X ∈ C, there is a morphism εX : I → X ⊗ X that satisﬁes the following conditions:

• For all morphisms f : X → Y, the following diagram commutes:

εX I / X ⊗ X∗

εY f ⊗IdX∗ ∗ ∗ Y ⊗ Y ∗/ Y ⊗ X IdY ⊗ f 28

• For all X ∈ C, the following composition is IdX∗ :

X∗ / X∗ O −1 ρ ∗ λX∗ X I ⊗ X∗ X∗ ⊗ I O −1 εX∗ ⊗IdX∗ IdX∗ ⊗ν (X∗ ⊗ X∗∗) ⊗ X∗ X∗ ⊗ I∗ O −1 ∗ α IdX∗ ⊗(εX) ∗ X∗ ⊗ (X∗∗ ⊗ X∗) / X∗⊗(X ⊗ X∗) IdX∗ ⊗γX∗,X

• For all X, Y ∈ C, the following composition is εX⊗Y :

∗ I / (X⊗Y)⊗(X⊗Y) O εX IdX⊗Y ⊗γY,X X⊗X∗ (X⊗Y)⊗(Y∗⊗X∗) O ⊗ −1 IdX λX∗ α X⊗(I⊗X∗) X ⊗ (Y ⊗ (Y∗⊗X∗)) 5

IdX⊗εY ⊗Id ∗ ⊗ −1 X ( IdX α X ⊗ ((Y⊗Y∗)⊗X∗)

Let C be a pivotal category, and let X ∈ C. The monoid EndC(X) = HomC(X, X) has two trace maps, TrL, TrR : EndC(X) → EndC(I), given by the following compositions:

TrL( f ) I / I O εX ν−1 X∗ ⊗ X∗∗ I∗ O ∗ −1 ∗ ∗ IdX ⊗τX ε (X ) X∗⊗X (X∗ ⊗ X∗∗)∗ O ∗∗ ∗ IdX∗ ⊗ f γX ,X X∗⊗X / X∗∗∗⊗X∗∗ τX∗ ⊗τX 29

TrR( f ) I / I O εX ν−1 X⊗X∗ I∗ O ∗ f ⊗IdX∗ εX X⊗X∗ (X⊗X∗)∗ 8

τ ⊗Id ∗ γX∗,X X X % X∗∗⊗X∗ In the case where C is strict, these compositions simplify to

∗ ∗ TrL( f ) = εX∗ (IdX∗ ⊗ f )εX∗ , TrR( f ) = εX( f ⊗ IdX∗ )εX .

A pivotal category C is called a spherical category if, for all objects X ∈ C and all morphisms f : X → X, we have TrL( f ) = TrR( f ). We denote the common value of the left and right traces simply by Tr( f ).

1.2 All About Algebras, or “So You Want to Multiply”

1.2.1 Algebras and Modules

Let k be a ﬁeld, and let C be a k-linear monoidal category. An associative unital algebra in C is an object A ∈ C together with two morphisms µ : A ⊗ A → A and ι : I → A such that

µ(µ ⊗ IdA) = µ(IdA ⊗ µ), and

µ(ι ⊗ IdA) = IdA = µ(IdA ⊗ ι).

Henceforth, when we refer to an “algebra” in a monoidal category, we will tacitly assume that the algebra is an associative unital algebra unless we explicitly state otherwise. Let A, B ∈ C be algebras. A right A-module is an object M ∈ C together with a 30 morphism ν : M ⊗ A → M such that

ν(ν ⊗ IdA) = ν(IdM ⊗ µ).

Left A-modules are deﬁned similarly. If M ∈ C is both a left A-module and a right B-module such that the respective actions are compatible, then M is said to be an (A, B)-bimodule. Compatibility of the actions amounts to commutativity of the following diagram:

IdA⊗ρ A ⊗ M ⊗ B / A ⊗ M

λ⊗IdB λ M ⊗ B ρ / M Here, ρ (resp. λ) denotes right (resp. left) action by B (resp. A) on M. We denote the

category of left A-modules in C by AC, that of right A-modules by CA, and that of

(A, B)-bimodules by ACB. An algebra A in a monoidal category C is said to be simple if any nonzero algebra homomorphism A → B is a monomorphism. An algebra A is said to be separable if it is equipped with a map ε : A → I such that the following composition is a nondegenerate pairing (denoted e : A ⊗ A → I):

µ ε A ⊗ A −→ A −→ I.

Nondegeneracy of e means that there is a morphism κ : I → A ⊗ A such that the composition Id ⊗κ e⊗Id A −→A A⊗A⊗A −→A A

is the identity. It also implies that the similar composition

κ⊗Id Id ⊗e A −→A A⊗A⊗A −→A A

is also the identity. 31

For a separable algebra A, the adjunction

+ C j CA

splits. Indeed, the splitting M → M ⊗ A of the adjunction map M ⊗ A → M is given by the composition

IdM⊗ε µ⊗IdA M / M ⊗ A⊗A / M ⊗ A .

An algebra A = hA, µ, ιi in a braided monoidal category is commutative if

µ = µ ◦ cA,A.

An indecomposable algebra A is an etale algebra if it is both commutative and separable. A right A-module M = hM, νi is said to be local if the following diagram commutes:

ν M ⊗ A / M (1.1) O cM,A ν A ⊗ M / M ⊗ A cA,M

Locality in this sense is unrelated to the notion of M having a unique maximal submodule.

loc We denote by CA the full subcategory of local right A-modules. An algebra L in a k-linear

loc braided monoidal category C is said to be a Lagrangian algebra if the category CL of local right L-modules is equivalent to the category k-Vect of k-vector spaces.

1.2.2 The Full Center of an Algebra

The full center34 Z(A) of an algebra A in a monoidal category C is an object of the monoidal center Z(C) together with a morphism Z(A) → A in C, terminal among pairs (Z, ζ), where Z ∈ Z(C) and ζ : Z → A is a morphism in C such that the following diagram

34 Cf.[9] and [12]. 32 commutes:

IdA⊗ζ A ⊗ Z / A ⊗ A (1.2) µ

" zA A <

µ Z ⊗ A / A ⊗ A ζ⊗IdA

Here, zA is the half-braiding of Z as an object of Z(C). The terminality condition means that for any such pair (Z, ζ), there is a unique morphism φ : Z → Z(A) in the monoidal center Z(C), which makes the following diagram commute:

φ Z / Z(A)

ζ ι A }

Here, ι : Z(A) → A denotes the canonical inclusion.

1.3 Group Cohomology

1.3.1 G-modules and the Standard Normalized Complex

Let G be a group, and let M be an Abelian group. We say that M is a G-module if G

acts Z-linearly on M, i.e., G acts on M such that

g.(m + n) = g.m + g.n ∀g ∈ G, m, n ∈ M.

Let M be a G-module. For an integer n ≥ 0, we denote by     n   C (G, M) = c : G × · · · × G → M c(g1,..., gn) = 0 if ∃ j s.t. g j = e  | {z }  n factors the group of normalized n-cochains. This is the Abelian group of set-theoretic functions

from the n-fold Cartesian power of G into M that vanish whenever at least one of their arguments is the identity element of G, with group operation given by pointwise addition. 33

For an integer n ≥ 0, the standard diﬀerential or coboundary homomorphism ∂n : Cn(G, M) → Cn+1(G, M) is given by the formula

n n X i [∂ (c)](g1,..., gn+1) = g1.c(g2,..., gn+1) + (−1) c(g1,..., gigi+1,..., gn+1)+ (1.3) i=1

n+1 + (−1) c(g1,..., gn).

Through an heroic piece of elementary algebra, it can be shown that

∂n+1 ◦ ∂n = 0 ∀n ≥ 0.

We deﬁne the group of normalized n-cocycles to be

Zn(G, M) = ker(∂n), and the subgroup of normalized n-coboundaries to be   {0} , n = 0   Bn(G, M) =  .   h i  ∂n−1 Cn−1(G, M) , n > 0

The nth cohomology group of G with coeﬃcients in M is the quotient group

Hn(G, M) = Zn(G, M) /Bn(G, M) .

Here, the condition that ∂n+1 ◦ ∂n = 0 is used to guarantee that Bn(G, M) ⊆ Zn(G, M), so that this quotient makes sense. 34

1.3.2 Abelian Cohomology

Let G be an Abelian group and M a trivial G-module. Under these assumptions, the standard diﬀerential (1.3) simpliﬁes to n n X i [∂ (c)](g1,..., gn+1) = c(g2,..., gn+1) + (−1) c(g1,..., gigi+1,..., gn+1)+ i=1 n+1 +(−1) c(g1,..., gn).

×n Let C∗(G) denote the graded Abelian group for which, for n > 0, Cn(G) = Z G , i.e.,

Cn(G) is the free Abelian group with basis {(g1, g2,..., gn) |gi ∈ G}. Deﬁne Z-linear maps dn+1 : Cn+1(G) → Cn(G) by n X n+1 dn+1(g1,..., gn+1) = (g2,..., gn+1) + (g1,..., gigi+1,..., gn+1) + (−1) (g1,..., gn). i=1

It can be shown that the maps dn are chain diﬀerentials.

0 An n, n -shuﬄe is a permutation σ ∈ S n+n0 such that σ(1) < σ(2) < ··· < σ(n) and σ(n + 1) < σ(n + 2) < ··· < σ n + n0. Denote by Shn, n0 the set of all n, n0-shuﬄes.

Deﬁne the shuﬄe product ? : Cn(G)⊗ZCn0 (G) → Cn+n0 (G) on generators by

X sgn(σ) (g1,..., gn) ? (gn+1,..., gn+n0 ) = (−1) gσ(1),..., gσ(n), gσ(n+1),..., gσ(n+n0) . σ∈Sh(n,n0) It can be shown that the shuﬄe product is associative and commutative, and satisﬁes the derivation identity

|x| d∗(x ? y) = d∗(x) ? y + (−1) x ? d∗(y), x, y ∈ C∗(G) .

The shuﬄe product endows C∗(G) with the structure of a diﬀerential graded algebra.

Denote by B∗(G) = B(C∗(G)) the bar complex of the diﬀerential graded algebra

C∗(G), i.e., for n > 0, Bn(G) is generated by symbols [x1| · · · |xr], where xi ∈ Cni (G) such

that ni > 0 and n1 + n2 + ··· + nr + (r − 1) = n. The diﬀerential d : Bn+1(G) → Bn(G) has the form Xr Xr−1 h i ai−1 a j d[x1| · · · |xr] = (−1) [x1| · · · |d (xi) | · · · |xr] + (−1) x1 ··· xi ? x j ··· xr , i=1 j=i 35 where ai = |x1| + |x2| + ··· + |xi| + i.

• The cohomology Hab(G, M) of the cochain complex HomZ(B∗(G), M) is called the Abelian cohomology. The ﬁrst Abelian cohomology group has the form

1 Hab(G, M) ' {φ : G → M |φ( f g) = φ( f ) + φ(g)} = HomZ(G, M) ,

and so it coincides with the ordinary ﬁrst cohomology group H1(G, M). The second Abelian cohomology group has the form

2 n 2 o . 2 Hab(G, M) = γ ∈ Z (G, M) γ( f, g) = γ(g, f ) ∀ f, g ∈ G B (G, M) ,

and hence is a subgroup of the ordinary second cohomology group H2(G, M) consisting of i.e. H2 (G, M) 1 (G, M) cohomology classes of symmetric 2-cocycles, , ab = ExtZ . The third Abelian cohomology group has the form

3 3 . 3 Hab(G, M) = Zab(G, M) Bab(G, M) ,

where an Abelian 3-cocycle is a pair (α, c) ∈ Z3(G, M) × C2(G, M) such that

• c( f, g) + c(g, h) + α( f, h, g) = c( f g, h) + α( f, g, h) + α(h, f, g), and

• c( f, h) + c( f, g) + α( f, g, h) = c( f, gh) + α(g, f, h) + α(g, h, f )

for all f, g, h ∈ G. An Abelian 3-cocycle (α, c) is an Abelian 3-coboundary if is of the form

(α, c) = (∂u( f, g, h), u( f, g) − u(g, f ))

for some u ∈ C2(G, M).

1.4 Categories, Part II, or “Generalized Abstract Nonsense”

1.4.1 Graded Vector Spaces

Let X be a set. A k-vector space V is X-graded if there exists a direct sum decomposition M V = Vx x∈X 36 of V into k-vector subspaces indexed by the set X. A G-graded vector space M A = Ag g∈G is said to be a G-graded algebra if it is equipped with a multiplication that preserves the G-grading:

A f Ag ⊆ A f g, f, g ∈ G.

For a 3-cocycle α ∈ Z3(G, k∗), a G-graded algebra A is α-associative if

a(bc) = α( f, g, h)(ab)c, ∀a ∈ A f , b ∈ Ag, c ∈ Ah.

A G-algebra is an algebra A ∈ k-Vect together with an action of G on A by algebra homomorphisms.

1.4.2 The Category V(G, α)

For a group G, a ﬁeld k, and a 3-cocycle α ∈ Z3(G, k∗), we deﬁne the monoidal category V(G, α) as follows. Objects of V(G, α) are G-graded k-vector spaces. Morphisms in V(G, α) are k-linear maps T : X → Y that preserve the G-grading in the h i sense that, for each g ∈ G, we have T Xg ⊆ Yg. The tensor product in V(G, α) is the usual tensor product of graded vector spaces, viz.,

M (V ⊗ W) f = Vg ⊗ Wh , f ∈ G, (1.4) g,h∈G gh= f

so that     M M  V ⊗ W =  Vg ⊗ Wh .   f ∈G g,h∈G  gh= f Associativity of the tensor product in V(G, α) is twisted by α in the following way. Let

U, V, W ∈ V(G, α), and assume that u ∈ U f , v ∈ Vg, and w ∈ Wh for f, g, h ∈ G. Then

u ⊗ (v ⊗ w) = α( f, g, h)(u ⊗ v) ⊗ w. (1.5) 37

The monoidal unit in V(G, α) is I = Ie = k.

1.4.3 The Category Z(G, α)

Let G be a group, and let α ∈ Z3(G, k∗) be a normalized 3-cocycle of G with coeﬃcients in the trivial G-module k∗. An α-projective G-action on a G-graded vector space V is a collection of automorphisms f : V → V for each f ∈ G such that h i f Vg = V f g f −1 , and

( f g).v = α( f, g|h) f. (g.v) , ∀v ∈ Vh. (1.6)

Here, α( f, ghg−1, g) α( f, g|h) = . (1.7) α( f, g, h)α( f ghg−1 f −1, f, g) Similarly, deﬁne α( f, g, h)α( f g f −1, f h f −1, f ) α( f |g, h) = . (1.8) α( f g f −1, f, h) The following identities follow directly from the 3-cocycle condition for normalized α:

α( f, gh|u)α(g, h|u) = α( f g, h|u)α( f, g|huh−1), (1.9)

α(g, h, u)α( f |gh, u)α( f |g, h) = α( f |g, hu)α( f |h, u)α( f g f −1, f h f −1, f u f −1).

The twisted Drinfeld center Z(G, α) is the braided monoidal category deﬁned as follows. Objects of Z(G, α) are G-graded vector spaces

M V = Vg g∈G

together with a given α-projective G-action. We will denote the action of an element g ∈ G on a vector v ∈ V by g.v. Morphisms of Z(G, α) are graded and action-preserving homomorphisms of vector spaces. The tensor product in Z(G, α) is the tensor product of 38

G-graded vector spaces as in (1.4), with associativity twisted by α as in (1.5), and α-projective G-action deﬁned by

f. (x ⊗ y) = α( f |g, h) ( f.x) ⊗ ( f.y) , x ∈ Xg, y ∈ Yh. (1.10)

The monoidal unit is I = Ie = k with trivial G-action. The braiding is given by

cX,Y (x ⊗ y) = ( f.y) ⊗ x, x ∈ X f , y ∈ Y. (1.11)

The category Z(G, α) is rigid35, with dual objects

∗ M ∗ X = (X ) f f ∈G given by

∗ ∗ (X ) f = X f −1 = Homk X f −1 , k , f ∈ G,

with the α-projective action

1 −1 (g.`) (x) = ` g .x , ` ∈ Hom X −1 , k , x ∈ X −1 −1 . α(g| f, f −1)α(g−1, g| f −1) k f g f g

The category Z(G, α) is unitarizable with the ribbon twist

−1 −1 θX(x) = α( f, f , f ) f .x, x ∈ X f .

The (unitary) trace of an endomorphism a : X → X can be written in terms of ordinary

traces on vector spaces Xg: X Tr(a) = TrXg ag , g∈G and the (unitary) dimension of an object X ∈ Z(G, α) is the dimension of its underlying (graded) vector space X dim(X) = dim Xg . g∈G It will be shown that the Drinfeld center Z(G, α) is precisely the monoidal center of the category V(G, α) of graded vector spaces.

35 See lemma 2.2.2.1. 39

1.4.4 The Category C(G, H, α)

For a (ﬁnite) group G, a subgroup H ≤ G, and a 3-cocycle α ∈ Z3(G, k∗), we deﬁne the monoidal category C(G, H, α) as follows: Objects of C(G, H, α) are G-graded k-vector spaces with a given α|H-projective H-action. Morphisms of C(G, H, α) are k-linear maps preserving both the G-grading and H-action. The tensor product in C(G, H, α) is the tensor product of G-graded k-vector spaces, with α|H-projective H-action given by

h.(x ⊗ y) = α(h| f, g)(h.x) ⊗ (h.y), x ∈ X f , y ∈ Yg, h ∈ H, f, g ∈ G.

The monoidal unit in C(G, H, α) is I = Ie = k with trivial H-action. Note that the category

C(G, H, α) deﬁned above contains, as a full subcategory, the braided category Z(H, α|H), and that the category C(G, G, α) is Z(G, α).

1.4.5 The Category C(A, q) and Quadratic Functions

Let A be an Abelian group, let α ∈ Z3(A, k∗) be normalized, and let c ∈ C2(A, k∗) be a

3 ∗ normalized 2-cochain such that the pair (α, c) ∈ Hab(A, k ) is an Abelian 3-cocycle. Deﬁne the pointed category C(A, α, c) as follows. Objects of C(A, α, c) are A-graded k-vector spaces. Morphisms of C(A, α, c) are linear maps preserving the A-grading. The category C(A, α, c) is braided monoidal with respect to the tensor product of graded vector spaces, with associativity given by

αV,U,W (v ⊗ (u ⊗ w)) = α(a, b, c)((v ⊗ u) ⊗ w), v ∈ Va, u ∈ Ub, w ∈ Wc, a, b, c ∈ A,

and braiding given by

cV,W (v ⊗ w) = c(a, b)(w ⊗ v), v ∈ Va, w ∈ Wb . (1.12)

The category C(A, α, c) is fusion, with its simple objects being one-dimensional vector spaces I(a) concentrated in degree a ∈ A. 40

Let A be an Abelian group. A function q : A → M is a quadratic function if the

2 following hold: For all n ∈ Z and x ∈ A, we have q(nx) = n (q(x)), and the function σ : A × A → M deﬁned by σ(x, y) = q(x + y) − q(x) − q(y) is a symmetric Z-bilinear form. We call σ the polarization of q. The set of such quadratic functions forms a group that we denote by Q(A, M). Via the assignment (α, c) 7→ q : G → M, where q(x) = c(x, x), the

3 Abelian cohomology group Hab(G, M) is isomorphic to Q(G, M). We apply the correspondence (α, c) 7→ q to write C(A, α, c) = C(A, q) for short. When the quadratic function q is trivial, we will write C(A) for C(A, q). The category

C(A) is symmetric and coincides with Rep(Ab). Here, Ab = HomZ(A, Q/Z). A quadratic function q : A → M is nondegenerate if its polarization σ is nondegenerate. In the case where A is ﬁnite, a quadratic function q : A → M is nondegenerate if and only if ker(σ) = {0}. For a subgroup B of a quadratic Abelian group (A, q), the orthogonal complement of B is the subgroup B⊥ ≤ A given by

⊥ B = a ∈ A q(a + b) = q(a) + q(b) ∀b ∈ B .

A subgroup B ≤ A is isotropic (with respect to q) if q|B ≡ 0. If B is an isotropic subgroup of A, then B ⊆ B⊥. In this case, we call the quotient group B⊥ B the isotropic contraction along B. The quadratic function q descends to the isotropic contraction B⊥ B. We call an isotropic subgroup L of a nondegenerate quadratic group (A, q) Lagrangian if L = L⊥. If A is ﬁnite, then L is Lagrangian if and only if |L|2 = |A|.

1.4.6 Categorical Groups and Their Standard Invariants

Following [26, § 3.1], we deﬁne a categorical group to be a monoidal category hG, ⊗, Ii in which all morphisms are invertible and, for every object X ∈ G, there is an

∗ ∗ object X ∈ G and a morphism eX : X ⊗X → I. 41

Let G be a categorical group. The standard invariants of G are the zeroth and ﬁrst homotopy groups π0(G) and π1(G), respectively. π0(G) is the group of isomorphism classes of objects of G, and π1(G) is the group of automorphisms of the monoidal unit I ∈ G. The

3 associativity constraint α for the tensor product in G is a class in H (π0(G), π1(G)). Thus up to equivalence, categorical groups are labeled by triples (π0(G), π1(G), [α]). Here,

3 [α] ∈ H (π0(G), π1(G)) is the cohomology class of the associator of G.

1.4.7 Module Categories

Let C be a monoidal category with associator α.A module category over C is a category M equipped with an action ∗ : C × M → M, a unit isomorphism

`M : Id⊗M → M, and a natural isomorphism aX,Y,M :(X⊗Y) ∗ M → X ∗ (Y ∗ M) for all X, Y ∈ C and M ∈ M such that the pentagon diagram

(X⊗(Y⊗Z)) ∗ M (1.13) 5 αX,Y,Z ⊗IdM aX,Y⊗Z,M

) ((X⊗Y)⊗Z) ∗ M X ∗ ((Y⊗Z) ∗ M)

aX⊗Y,Z,M IdX⊗aY,Z,M (X⊗Y) ∗ (Z ∗ M) X ∗ (Y ∗ (Z ∗ M)) 7

aX,Y,Z∗M commutes for all X, Y, Z ∈ C and M ∈ M. An action of C on M is nothing but a monoidal functor C → End(M).

1.4.8 Module Functors

Let M and N be module categories with associativity constraints m and n, respectively. Then a module functor F : M → N is an isomorphism

FX,M : F(X ∗ M) → X ∗ F(M) for all X ∈ C and M ∈ M such that the following diagrams 42 commute: F(X ∗ (Y ∗ M)) (1.14) 5 F(mX,Y,M) FX,Y∗M

) F((X ⊗ Y) ∗ M) X ∗ F(Y ∗ M)

FX⊗Y,M IdX∗FY,M (X ⊗ Y) ∗ F(M) X ∗ (Y ∗ F(M)) 7

nX,Y,F(M) and

F(Id⊗M) F(Id⊗M) / Id⊗F(M)

F(` ) ` m % y F(M) F(M) for all X, Y ∈ C and M ∈ M. For module functors F : M → N and G : N → P, the composition G ◦ F has the following module structure:

G(FV,M) GV,F(M) G(F(V ∗ M) −−−−−→ G(V ∗ F(M)) −−−−−→ V ∗ G(F(M))

1.4.9 Module Natural Transformations

Now let F and G be functors of C-module categories M → N. Let a be a natural transformation of F into G. Then a is given by a collection of isomorphisms aM : F(M) → G(M), M ∈ M, such that the following diagram commutes:

FX,M F(X ∗ M) / X ∗ F(M) (1.15)

aX∗M IdX∗aM G(X ∗ M) / X ∗ G(M) GX,M

1.4.10 Simple V(G, α)-module Categories

Consider a pointed fusion category of the form C = V(G, α), where G is a ﬁnite group. Let X be a G-set. Deﬁne an action of V(G, α) on the category V(X) of X-graded 43 vector spaces by

M (V ∗ M)x = Vg⊗Mx, V ∈ V(G, α), M ∈ V(X) . g.y=x

For γ ∈ C2(G, Map(X, k∗)), deﬁne the action associativity by

aU,V,M((u⊗v) ∗ m) = γ( f, g, x)u ∗ (v ∗ m), u ∈ U f , v ∈ Vg, m ∈ Mx .

The pentagon axiom (1.13) is equivalent to the condition that ∂γ = α. We will denote the resulting V(G, α)-module category by M(X, γ). For this example, we can generate full module functors F = F( f, ϕ): M(X, γ) → M(Y, η) using the following choices. First, choose a map of G-sets f : X → Y and ϕ : G × X → k∗ with ϕ ∈ C1(G, Map(X, k∗)). Coherence of (1.14) implies γ = ∂(ϕ) f ∗(η) while the deﬁnition gives for M ∈ M,

M F( f, ϕ)(M) = F(M)y = Mx f (x)=y The composition of module functors F( f, ϕ) and F0( f 0, ϕ0) is

F( f, ϕ) ◦ F0( f 0, ϕ0) = F( f 0 f, f ∗(ϕ0)ϕ) .

For g ∈ Vg and x ∈ X, the composition has the form

F0(F(g, x)) = F0( f (g), ϕ(g, x)) = ( f 0( f (g)), ϕ0(g, f (x))) .

In the speciﬁc case where M = N = M(X, γ), and F = F( f, ϕ) and G = F( f 0, ϕ0), a natural transformation a is an element of Map(X, k∗) = C0(G, Map(X, k∗)), and the commutativity of (1.15) amounts to the equality f (a) · ϕ = ϕ0. A V(G, α)-module category V(X, γ) is simple if and only if X is a transitive G-set. Write X = G/H for some subgroup H ≤ G. The associativity constraint γ is a class in C2(G, Map(G/H, k∗)). Simple V(G, α)-module categories correspond to pairs (H, γ),

2 ∗ where H ≤ G is a subgroup and γ ∈ C (H, k ) is a coboundary ∂γ = α|H. 44 2 Preliminary Results

Here, we establish a number of general preliminary results that will be of use later.

2.1 Algebras in Monoidal Categories

2.1.1 Maschke’s Lemma

Here, we state and prove a well-known result of classical representation theory.

Lemma 2.1.1.1 ([42], Proposition 9; [24], Proposition IX.5.8). Let k be a ﬁeld, and let G

be a ﬁnite group. The group algebra k[G] is semisimple if and only if char(k) - |G|. In particular, if char(k) = 0, then the group algebra k[G] is semisimple.

Proof. (⇐) Suppose that char(k) - |G|. Let M be a left k[G]-module, and let N ⊆ M be a submodule. Then in particular, N is a k-vector subspace of M, so there is a complementary subspace W such that M = N ⊕ W. Hence every element m ∈ M can be written uniquely as m = n + w for some n ∈ N, w ∈ W. Deﬁne π : M → N by π(n + w) = n. Observe that π is a k-linear projection of M onto N, and that ker(π) = W.

Since char(k) - |G|, either char(k) = 0 or |G| is a unit modulo char(k). In view of this fact, deﬁneπ ˆ : M → N by 1 X πˆ(m) = gπ g−1m . |G| g∈G Note thatπ ˆ is k-linear, ker(πˆ) = W, andπ ˆ[M] ⊆ N. We claim thatπ ˆ is k[G]-linear. Given m ∈ M and x ∈ G, we have

1 X πˆ(xm) = gπ g−1 xm . |G| g∈G

As g runs over G, so does h := x−1g, whence we have   1 X  1 X  πˆ(xm) = xhπ h−1m = x hπ h−1m  = xπˆ(m) . |G| |G|  h∈G h∈G Thereforeπ ˆ is k[G]-linear, as claimed. 45

To show thatπ ˆ is a projection onto N, it suﬃces to show thatπ ˆ 2 = πˆ. Observe that, given n ∈ N and g ∈ G, we have (gn) ∈ N, whence π(gn) = gn. Therefore 1 X 1 X 1 X πˆ(n) = gπ g−1n = g g−1n = gg−1 n = |G| |G| |G| g∈G g∈G g∈G 1 X 1 = n = · |G| · n = n . |G| |G| g∈G 2 It follows thatπ ˆ|N = IdN, and thereforeπ ˆ = πˆ, so thatπ ˆ is a projection with image N. It follows that M = N ⊕ W is a direct sum of k[G]-modules, whence k[G] is semisimple. (⇒) By contrapositive. Suppose that char(k) | |G|. Note that in this case, we must have X char(k) > 0. Let w := g. Note that w , 0. Moreover, for h ∈ G, we have g∈G   X  X hwh−1 = h gh−1 = hgh−1 ,   g∈G g∈G which is equal to w because conjugation is an automorphism of G. It follows that hw = wh for all h ∈ G, whence w ∈ Z(k[G]). Let n := |G|, and enumerate G as G = {g1, g2,..., gn}. Then  n   n  n   n  X  X  X  X  w2 =  g   g  = g  g  .  i  j  i j i=1 j=1 i=1 j=1 Xn Since for each i = 1, 2,..., n we have gi g j = giw = w, it follows that j=1 Xn w2 = w = nw . i=1 Since char(k) | n, it follows that w2 = 0. Let I := k[G]w. In view of the previous argument, I2 = {0}, and hence I is nilpotent, so it follows that I ⊆ J (k[G]). Moreover, w ∈ I and w , 0, so I , {0}. Therefore J(k[G]) , {0}, whence k[G] is not semisimple.

Remark 2.1.1.2. In what follows, we will be more concerned with the case where the regular group algebra k[G] is replaced by a twisted group algebra of the form kG, β for

2 ∗ some β ∈ Z (G, k ). It is still true in this case that, if char(k) - |G|, then the twisted group algebra kG, β is semisimple. 46

2.1.2 The Simple Things

Lemma 2.1.2.1 ([12], Lemma 2.1). Let A be an indecomposable separable algebra in a spherical fusion category C. Then A is simple.

Proof. If a separable algebra A is not simple, then there is a surjective, but not bijective, algebra homomorphism A B. Via the inverse image functor, the category of right

B-modules CB becomes a full subcategory of CA. Moreover, CB is a full left C-module

subcategory of CA. Recall from [18, Proposition 3.9] that a semisimple module category

over a fusion category is a direct sum of its simple module subcategories. In particular, CB is a direct summand of CA as a C-module category. Hence the algebra HomC(I, A) in k-Vect, which coincides with the algebra EndC IdCA of C-module endomorphisms of the identity functor of CA, is a nontrivial direct sum. Thus A is decomposable.

2.1.3 The Full Center of an Algebra

Here, we enumerate several properties of the full center36 of an algebra.

Proposition 2.1.3.1 ([12], Proposition 2.3). The full center Z(A) of an indecomposable separable algebra A in a fusion category C coincides with the action internal end

[A, A] ∈ Z(C) of the trivial bimodule A ∈ ACA with respect to the Z(C)-action on ACA given by α-induction. Moreover, the category of modules Z(C)Z(A) is equivalent, as a fusion category, to the category ACA of (A, A)-bimodules.

Proof. The universal property of the action internal end says that [A, A] is the terminal object among pairs (Z, z) where Z ∈ Z(C) and ξ : α(Z) → A is a morphism of (A, A)-bimodules. The right A-module map ξ : Z⊗A → A is completely determined by the morphism ζ = ξ(Id⊗ι): Z → A (which is still a left A-module map). The left A-module property for ζ is precisely (1.2).

36 Cf. [9], [12], and the diagram (1.2). 47

According to [38], the functors

−⊗[A,A]A ' Z(C)[A,A] ACA h [A,−] are quasi-inverse equivalences. The tensor structure

[A, M]⊗[A,A][A, N] → [A, M⊗AN] (2.1) for the functor [A, −] comes from the universal property of the action internal hom. Indeed, the composition

α([A, M]⊗[A, N])⊗AA ' α([A, M])⊗AA⊗Aα([A, N])⊗AA −→ M⊗AN

induces the map [A, M]⊗[A, N] → [A, M⊗AN], which is naturally [A, A]-balanced, viz., factors through [A, M]⊗[A,A][A, N], giving rise to (2.1).

Remark 2.1.3.2. Here is a slightly diﬀerent proof of the second statement of proposition 2.1.3.1, due to A. Davydov. The canonical braided equivalence (Morita invariance of the monoidal center) Z(C) → Z(ACA) sends the full center Z(A) to the full center Z(I) of the monoidal unit I ∈ ACA (which is really the (A, A)-bimodule A).

For a fusion category D (= ACA), the full center Z(I) ∈ Z(D) coincides with L(I), where L : D → Z(D) is the adjoint to the forgetful functor F : Z(D) → D. The adjunction is monadic. Moreover, the monad T = L ◦ F on Z(D) is a Z(D)-module functor. Thus T-algebras are the same as T(I)-modules. Finally, T(I) = L(I) = Z(I), and the forgetful functor factorizes:

Z(D) F

∼ # Z(D)Z(I) / D Theorem 2.1.3.3 ([12], Theorem 2.5). The full center Z(A) of an indecomposable separable algebra A in a fusion category C is a Lagrangian algebra in Z(C). 48

Proof. The tensor equivalence Z(C)Z(A) → ACA from proposition 2.1.3.1 induces a braided tensor equivalence Z Z(C)Z(A) → Z(ACA). By Morita invariance of the

monoidal center, Z(ACA) 'Z(C). By [16, Proposition 3.7], we have the decomposition into the Deligne product

loc Z(C) Z(C)Z(A) 'Z Z(C)Z(A) .

loc Combining the above, we get Z(C) Z(C)Z(A) 'Z(C), which means that loc Z(C)Z(A) ' k-Vect, i.e., Z(A) is Lagrangian.

2.2 Commutative Algebras in Group-theoretical Categories

2.2.1 Group-theoretical Braided Fusion Categories

In what follows, we will frequently make use of the following result, which was established in [19]. We state it here without proof.

Lemma 2.2.1.1 ([19]). Let (A, m, i) be a commutative algebra in a braided category C.

Let (B, µ, ι) be an algebra in CA. Deﬁne µ and ι as compositions

µ i ι B ⊗ B −→ B ⊗A B −→ B, I −→ A −→ B.

Then (B, µ, ι) is an algebra in C. Moreover, the map ι : A → B is a homomorphism of algebras in C. Furthermore, the algebra (B, µ, ι) in C is separable or commutative if and

locloc loc only if the algebra (B, µ, ι) in CA is such. Finally, the functor C −→ C given by A B B

m (M, m : B ⊗A M → M) 7→ M, m : B ⊗ M → B ⊗A M −→ M (2.2)

is a braided monoidal equivalence.

2.2.2 Ribbon Structure on Group-theoretical Braided Fusion Categories

Lemma 2.2.2.1 ([12], Lemma 3.2). The category Z(G, α) is rigid, with dual objects

∗ M ∗ X = (X ) f f ∈G 49 given by

∗ ∗ (X ) f = X f −1 = Homk X f −1 , k ,

with the α-projective action

1 −1 (g.`) (x) = ` g .x , ` ∈ Hom X −1 , k , x ∈ X −1 −1 . αg| f, f −1αg−1, g| f −1 k f g f g

Proof. Using the identities (1.9), it can be shown that the formula given above does indeed deﬁne an α-projective action on the associated graded vector space. The evaluation

∗ morphism evX : X ⊗ X → I is given by the canonical pairing

∗ (X ) f ⊗ X f −1 → k.

∗ Similarly, the coevaluation morphism coevX : I → X ⊗ X is given by the canonical elements

∗ k → X f −1 ⊗ (X ) f .

The category Z(G, α) is ribbon, with the ribbon twist

−1 −1 θX(x) = α f, f , f f .x, x ∈ X f .

The (unitary) trace of an endomorphism a : X → X can be written in terms of ordinary

traces on vector spaces Xg: X Tr(a) = TrXg ag , g∈G and the (unitary) dimension of an object X ∈ Z(G, α) is the dimension of its underlying (graded) vector space X dim(X) = dim Xg . g∈G 50

2.2.3 The Monoidal Center of V(G, α)

Here, we describe the monoidal center Z(V(G, α)) of the category of graded vector spaces. We begin with a simple lemma.

Lemma 2.2.3.1. Let α ∈ Z3(G, k∗) be a normalized 3-cocycle. Then we have the following identities:

α( f, g, h)γ(g, h)γ( f, gh) = α(h−1, g−1, f −1)γ( f, g)γ( f g, h), (2.3)

τ( f g)τ( f )−1τ(g)−1 = γ(g−1, f −1)γ( f, g)−1, (2.4)

where γ( f, g) = α( f g, g−1, f −1)−1α( f, g, g−1) and τ( f ) = α( f −1, f, f −1) = α( f, f −1, f )−1.

Proof. Indeed, for a normalized 3-cocycle α, the ratio of left- and right-hand sides of (2.3) is equal to ∂α( f gh, h−1, g−1, f −1)∂α( f g, h, h−1, g−1) , ∂α( f, g, h, h−1g−1)∂α(g, h, h−1, g−1) while the ratio of left- and right-hand sides of (2.4) coincides with

∂α( f, g, f −1g−1)∂α( f, g, ( f g)−1, f g)∂α( f, f −1, f, g) . ∂α(g−1, f −1, f, g)∂α(g, g−1, f −1, f g)

It is well-known37 that associativity constraints for the category of G-graded vector spaces with the ordinary tensor product (1.4) correspond to 3-cocycles

φV,U,W : V ⊗ (U ⊗ W) → (V ⊗ U) ⊗ W,

φV,U,W (v ⊗ (u ⊗ w)) = α( f, g, h)(v ⊗ u) ⊗ w, ∀v ∈ V f , u ∈ Ug, w ∈ Wh.

Denote by V(G, α) such a monoidal category. Clearly, V(G, α) is a fusion category with the set of simple objects Irr(V(G, α)) = G. It is quite straightforward to verify that the category of ﬁnite-dimensional vector

∗ spaces is spherical, with dual objects V = Homk(V, k) given by spaces of linear maps

37 See, e.g.,[26]. 51

−1 ∗ ∗ ∗ (algebraic dual spaces), with the inverse monoidal structure γV,U : U ⊗ V → (U ⊗ V) for the functor (·)∗ deﬁned as

−1 ∗ ∗ γV,U (m ⊗ `) = (m ⊗ `), ` ∈ V , m ∈ U ,

∗∗ where (m ⊗ `)(x ⊗ y) = `(x)m(y), and with the involution isomorphism τV : V → V given by

∗ τV (v) = x, x(`) = `(x), ` ∈ V .

This structure passes without change to the category of graded vector spaces with trivial associativity. The introduction of nontrivial coeﬃcients in associativity constraints perturbs this structure in the following way.

Proposition 2.2.3.2. The fusion category V(G, α) is spherical, with dual objects

∗ M ∗ X = (X ) f f ∈G given by

∗ ∗ (X ) f = X f −1 = Homk X f −1 , k ,

−1 ∗ ∗ ∗ ∗ with the inverse monoidal structure γX,Y : Y ⊗ Y → (X ⊗ Y) for the functor (·) deﬁned as 1 γ−1 (m ⊗ `) = · (m ⊗ `), ` ∈ X∗, m ∈ Y∗, X,Y γ( f, g) f g

∗∗ and with the involution isomorphism τX : X → X , given by

τX(x) = τ( f )x, x ∈ X f .

Here, γ( f, g) and τ( f ) are the same as in lemma 2.2.3.1.

Proof. Coherence for the monoidal structure γ

γX,YZ γY,Z IdX∗ (X ⊗ (Y ⊗ Z))∗ / (Y ⊗ Z)∗ ⊗ X∗ / (Z∗ ⊗ Y∗) ⊗ X∗ O O ∗ ∗ ∗ ∗ αX,Y,Z αZ ,Y ,X ∗ ∗ ∗ ∗ ∗ ∗ ((X ⊗ Y) ⊗ Z) γ / Z ⊗ (X ⊗ Y) / Z ⊗ (Y ⊗ X ) XY,Z IdZ∗ γX,Y 52 is equivalent to the equation (2.3), while monoidality of the isomorphism τ

∗ γX,Y (X ⊗ Y)∗∗ o (Y∗ ⊗ X∗)∗ O τX,Y γY∗,X∗ X ⊗ Y / X∗∗ ⊗ Y∗∗ τXτY

is equivalent to the equation (2.4).

Proposition 2.2.3.3 ([12], Proposition 4.1). The monoidal center Z (V(G, α)) is isomorphic, as braided monoidal category, to the category Z(G, α). The canonical forgetful functor Z (V(G, α)) → V(G, α) corresponds to the functor Z(G, α) → V(G, α) forgetting the G-action.

Proof. For an object (X, x) of the monoidal center Z(V(G, α)), the natural isomorphism

xV : V ⊗ X → X ⊗ V, V ∈ V(G, α)

is deﬁned by its evaluations on one-dimensional graded vector spaces. The isomorphism

xI( f ) can be seen as an automorphism f : X → X. The fact that xI( f ) preserves grading, h i amounts to the condition f Xg = X f g f −1 :

xI( f ) Xg = (I( f ) ⊗ X) f g / (X ⊗ I( f )) f g = X f g f −1 .

The coherence condition for x is equivalent to the the equation (1.6), with the associativity constraints giving rise to α(h| f, g). The diagram, deﬁning the second component x|y of the tensor product (X, x) ⊗ (Y, y) = (X ⊗ Y, x|y), is equivalent to the tensor product of projective actions (1.10), with the associativity constraints giving rise to α(g, h| f ). The description of the monoidal unit in a monoidal center corresponds to the answer

for the monoidal unit in Z(G, α). Clearly, the braiding c(X,x),(Y,y) = yX of the monoidal

center Z(V(G, α)) corresponds to the braiding (1.11) of Z(G, α). 53 3 Lagrangian Algebras in Group-theoretical Braided

Fusion Categories

In this chapter, we study Lagrangian algebras in the Drinfeld center Z(G, α) of a ﬁnite group G. We show that such algebras in Z(G, α) are parametrized by pairs (H, γ), where H is a subgroup of G and γ is a 2-cochain of H with values in k∗ that trivializes α in the sense that the diﬀerential of γ is α on H.

3.1 Description of the Category Z(G, α)

3.1.1 Simple Objects of Z(G, α)

Lemma 3.1.1.1. The category Z(G, α) admits the following decomposition into a direct sum of k-linear subcategories: M Z(G, α) = Z f (G, α), (3.1) f ∈c`(G) where the sum is taken over a set c`(G) of representatives of conjugacy classes of elements of G, and for f ∈ G, the subcategory Z f (G, α) is given by

Z f (G, α) = Z ∈ Z(G, α) supp(Z) = c`( f ) .

n − o Here, c`( f ) = g f g 1 g ∈ G denotes the conjugacy class of f in G.

Proof. Clearly, the support of an object of Z(G, α) is a union of conjugacy classes of G. It is also straightforward that, for Z ∈ Z(G, α), we have M Z = Zc , c∈c`(G) where M Zc = Z f , f ∈c giving a decomposition of Z into a direct sum of objects of Z(G, α). The result follows. 54

Lemma 3.1.1.2. For f ∈ G, the category Z f (G, α) is equivalent, as a k-linear category, to

h −1i the category k CG( f ), α(·, ·| f ) -Mod of left modules over the twisted group algebra

h −1i k CG( f ), α(·, ·| f ) in the category k-Vect of vector spaces.

h −1i Proof. Consider the assignment F : Z f (G, α) → k CG( f ), α(·, ·| f ) -Mod given by

h −1i F(Z) = Z f . We make F(Z) into a left k CG( f ), α(·, ·| f ) -module by deﬁning, for

h ∈ CG ( f ) and z ∈ Z f , eh · z = h.z. It can be easily checked that this deﬁnition gives rise to

h −1i a left k CG( f ), α(·, ·| f ) -module structure on F(Z).

h −1i Now consider the assignment E : k CG( f ), α(·, ·| f ) -Mod → Z f (G, α) given by

−1 −1 −1 −1 E(M) = a : G → M a(xy) = α y , x |x f x y .a(x) ∀x ∈ G, y ∈ CG( f ) .

Deﬁne G-grading on E(M) as follows: a function a ∈ E(M) is homogeneous if and only if

supp(a) is a single coset modulo CG ( f ), i.e.,

−1 |a| = x f x ⇐⇒ supp(a) = xCG( f ).

In particular,

|a| = f ⇐⇒ supp(a) = CG( f ). (3.2)

− − − Deﬁne α-projective G-action on E(M) by (g.a) (x) = α x 1, g x f x 1 a g 1 x . It can be

checked that these deﬁnitions make E(M) into an object of Z f (G, α).

Now we proceed to deﬁning the adjunction isomorphisms. For Z ∈ Z f (G, α), given a

homogeneous z ∈ Zx f x−1 for some x ∈ G, deﬁnez ˜ : G → Z f by   −1  y .z, y ∈ xCG( f ) z˜(y) =  .   0, otherwise

Observe that, for g ∈ CG( f ), we have

− − − − − − − − − − − z˜(xg) = g 1 x 1 .z = α g 1, x 1 x f x 1 g 1. x 1.z = α g 1, x 1 x f x 1 g 1.(z˜(x)), 55 so thatz ˜ ∈ E(F(Z)). Now deﬁne φZ : Z → E(F(Z)) by φZ(z) = z˜. We ﬁrst check that φZ preserves the α-projective action. We have

1 φ (g.z)(x) = g.z(x) = x−1.(g.z) = x−1g. .z = Z f α(x−1, g|x f x−1) 1 = z˜ g−1 x = (g.z˜)(x) = (g.φ (z))(x), α(x−1, g|x f x−1) Z and so φZ preserves the α-projective action. Now we check that φZ preserves the grading. Let |z| = b f b−1 for some b ∈ G. Recall from equation (3.10) that

|z˜| = b f b−1 ⇐⇒ y|z˜(y)|y−1 = b f b−1 ∀y ∈ G.

Since supp(z˜) = xCG( f ), it suﬃces to check that this holds when y = xg for some g ∈ CG( f ). We obtain

− − − − − (yg)|z˜(yg)|(yg) 1 = (yg) (yg) 1.z (yg) 1 = (yg)(yg) 1|z|(yg)(yg) 1 =

= e|z|e = |z| = b f b−1, as claimed. Therefore φZ preserves the grading, whence φZ is a morphism in the category

Z f (G, α). To see that φZ is an isomorphism, we exhibit an inverse. Deﬁne a map

ηZ : E(F(Z)) → Z by

1 X η (z) = α g, g−1 |a| g. (a(g)). (3.3) Z |C ( f )| G g∈G

−1 −1 We claim that ηZ = φZ . Let z ∈ Zx f x−1 for some x ∈ G, so that |z˜| = x f x as well. We have

1 X (η ◦ φ ) (z) = η (z˜) = α g, g−1 |z˜| g. (z˜(g)) = Z Z Z |C ( f )| G g∈G −1 α g, g |z˜| 1 X −1 −1 1 X = α g, g |z˜| g. g .z = z = |CG( f )| |CG( f )| −1 g∈xCG( f ) g∈xCG( f ) α g, g |z| 1 X 1 1 = z = |xCG( f )| · z = |CG( f )| · z = z, |CG( f )| |CG( f )| |CG( f )| g∈xCG( f ) 56 so that ηZ ◦ φZ = IdZ. Now let a ∈ E(F(Z)) be homogeneous of degree |a| = y f y−1, so that supp(a) = yCG( f ). For x ∈ supp(a), we have    X   1 −1  ((φZ ◦ ηZ) (a)) (x) = φZ  α g, g |a| g. (a(g)) (x) = |C ( f )|  G g∈G 1 X −1 = α g, g |a| g^.(a(g))(x) = |CG( f )| g∈yCG( f ) 1 X −1 −1 = α g, g |a| x . (g. (a(g))) = |CG( f )| g∈yCG( f ) α g, g−1 |a| 1 X −1 = x g . (a(g)) = |CG( f )| −1 g∈yCG( f ) α x , g |a(g)|

− 1 X −1 −1 −1 1 = α x g, g |a| g x . (a(g)). (3.4) |CG( f )| g∈yCG( f ) Set h = g−1 x, so that x = gh. Then we have that (3.4) coincides with

1 X −1 −1 −1 = α h , g |a| h . (a(g)). (3.5) |CG( f )| g∈yCG( f )

Since x ∈ supp(a) = yCG( f ) and g ∈ yCG( f ), it follows that gCG( f ) = xCG( f ), and

−1 therefore h = g x ∈ CG( f ). We apply CG( f )-equivariance of a to rewrite (3.4) as

1 X 1 a(x) = |yCG( f )| · a(x) = a(x), (3.6) |CG( f )| |CG( f )| g∈yCG( f )

−1 which shows that φZ ◦ ηZ = IdE(F(Z)), whence ηZ = φZ .

For an object V ∈ C ({ f }, CG( f ), α), we have F (E (V)) = (E(V)) f , so by equation

(3.2), a function a : G → V belongs to F (E(V)) if and only if supp (a) = CG( f ).

Moreover, the equivariance condition can be rewritten as follows: for x ∈ CG( f ), we have

− − − − a(x) = a(ex) = α e 1, x 1 f x 1.(a(e)) = x 1.(a(e)), (3.7) 57 where the last equality follows from the fact that α is normalized. Eﬀectively, therefore, any such function is completely determined by its value at the identity element e ∈ G. For

an object V ∈ C ({ f }, CG( f ), α), deﬁne ψV : F (E(V)) → V by

ψV (a) = a(e),

and deﬁne ξV : V → F(E(V)) by

ξV (v) = v˜, wherev ˜ : G → V is deﬁned by v˜(x) = x−1.v.

We ﬁrst check that ψV is a morphism in C ({ f }, CG( f ), α). Being an evaluation map, ψV is k-linear. For a ∈ F(E(V)), we have |a| = f , i.e.,

|a(x)| = x−1 f x ∀x ∈ G.

−1 Therefore |ψV (a)| = |a(e)| = e |a|e = |a| = f , so that ψV preserves the { f }-grading. Next

we check that ψV preserves the α-projective CG( f )-action. Let g ∈ CG( f ). Then

−1 ψV (g.a) = (g.a) (e) = a g = g. (a(e)) by CG( f )-equivariance, which coincides with g. (ψV (a)). Therefore ψV preserves the α-projective CG( f )-action, whence ψV is indeed a

−1 morphism in C ({ f }, CG( f ), α). Finally, we claim that ξV = ψV . We have

(ψV ◦ ξV ) (v) = ψF(E(V) (v˜) = v˜(e) = e.v = v.

On the other hand, for x ∈ CG( f ), we have

−1 ((ξV ◦ ψV ) (a)) (x) = (ξV (a(e))) (x) = ag(e)(x) = x . (a(e)) = a(x) in view of (3.7), and our claim is proven.

Corollary 3.1.1.3. Simple objects Z ∈ Z(G, α) are parametrized by conjugacy classes of pairs ( f, U), where f ∈ G and U is a simple module over the twisted group algebra

h −1i k CG( f ), α(·, ·| f ) . 58

Proof. Clearly, the support of a simple object Z ∈ Z(G, α) must be an indecomposable G-subset in G with respect to the adjoint action, i.e., a conjugacy class of G. Let g ∈ supp(Z). The axioms of the α-projective action imply that Z is induced from the

h −1i k CG(g), α(·, ·|g) -module Zg. For Z to be simple, the module Zg must be simple as well.

Setting f = g and U = Zg gives the result.

Corollary 3.1.1.4. If k is an algebraically closed ﬁeld of characteristic zero, then the category Z(G, α) is fusion.

Proof. Follows from corollary 3.1.1.3 and lemma 2.1.1.1.

3.2 Algebras in Group-theoretical Modular Categories

3.2.1 Algebras in the Category Z(G, α)

We start with expanding the structure of an algebra in the category Z(G, α) in plain algebraic terms.

Proposition 3.2.1.1 ([12], Proposition 3.3). An algebra A in the category Z(G, α) is a G-graded α-associative algebra together with an α-projective G-action such that

f.(ab) = α( f |g, h) ( f.a)( f.b) , a ∈ Ag, b ∈ Ah. (3.8)

An algebra A in the category Z(G, α) is commutative if and only if

ab = ( f.b) a, ∀a ∈ A f , b ∈ A. (3.9)

Proof. Being a morphism in the category Z(G, α), the multiplication of an algebra in Z(G, α) preserves grading and α-projective G-action, whence the property (3.8) holds. Associativity of multiplication in Z(G, α) is equivalent to α-associativity. The formula (1.11) for the braiding in Z(G, α) implies that commutativity for an algebra A in the

category Z(G, α) is equivalent to the condition (3.9). 59

Proposition 3.2.1.2 ([12], Corollary 3.4). The degree e component Ae of an algebra A in

the category Z(G, α) is an associative G-algebra and A is an (Ae, Ae)-bimodule. The

algebra Ae is commutative if A is a commutative algebra in the category Z(G, α).

Proof. The normalization condition for the cocycle α together with α-associativity of A

implies that Ae is an associative algebra and A is an (Ae, Ae)-bimodule. The same normalization condition together with α-projectivity of the G-action and the property (3.8)

implies that the action of G on Ae is genuine and G acts on Ae by algebra

homomorphisms. Commutativity of Ae for a commutative algebra A ∈ Z(G, α) follows

directly from the identity (3.9).

Proposition 3.2.1.3 ([12], Proposition 3.5). An algebra A in the category Z(G, α) is separable if and only if the composition

µ Tr A f ⊗ A f −1 −→ Ae −→ k

deﬁnes a nondegenerate bilinear pairing for any f ∈ G. In particular, the algebra Ae is separable if A is a separable algebra in the category Z(G, α).

Proof. Being a homomorphism of graded vector spaces, the standard trace map

Tr : A → I is zero on A f for f , e. Hence the standard bilinear form is zero on A f ⊗ Ag

unless f g = e. In particular, the restriction of Tr to Ae makes Ae into a separable algebra in

the category of vector spaces.

3.2.2 Commutative Separable Algebras in Trivial Degree and Their Modules

We start with a well-known38 description of etale G-algebras. We give a sketch of the proof for completeness.

38 See, e.g.,[29]. 60

Lemma 3.2.2.1 ([29], Theorems 2.1 and 2.2; [12], Lemma 3.6). Commutative separable G-algebras are function algebras on G-sets. Indecomposable G-algebras correspond to transitive G-sets.

Proof. An etale algebra over an algebraically closed ﬁeld k is a function algebra k(X) on a ﬁnite set X, with elements of X corresponding to minimal idempotents of the algebra. The G-action on the algebra amounts to a G-action on the set X. Clearly, the algebra of functions k(X t Y) on the disjoint union of G-sets is the direct sum of G-algebras

k(X) ⊕ k(Y), and any direct sum decomposition of G-algebras appears in that way.

Let k(X) be an indecomposable G-algebra. By choosing a minimal idempotent p ∈ X, we can identify the G-set X with the set G/H of left cosets modulo the stabilizer subgroup

H = stabG(p). This brings us to our ﬁrst major result. A sketch of the proof of this result was given in [12]; we provide a complete proof here.

Theorem 3.2.2.2 ([12], Theorem 3.7). Let G be a ﬁnite group, and let H ≤ G be a subgroup. Then the category Z(G, α)k(G/H) of right modules over the function algebra k(G/H) in the Drinfeld center Z(G, α) is equivalent, as a monoidal category, to the

loc category C(G, H, α). Moreover, the full subcategory Z(G, α)k(G/H) of local modules is

equivalent, as a braided monoidal category, to the Drinfeld center Z(H, α|H).

Proof. We shall exhibit the claimed equivalence of categories by constructing quasi-inverse functors D and E ﬁtting in the following diagram:

D ( Z(G, α)k(G/H) C(G, H, α) . h E Let p : G/H → k be deﬁned by    1, x ∈ H p(xH) =  ,   0, otherwise 61 so that p is, eﬀectively, the indicator function on H. Then p is a minimal idempotent in the function algebra k(G/H), and H = stabG(p). Let M be a right k(G/H)-module, and consider the assignment D : Z(G, α)k(G/H) → C(G, H, α) given by D(M) = Mp. Since p is of trivial degree, Mp is a G-graded vector space in a natural way: (Mp)g = Mg p. The G-action on M reduces to an H-action on Mp. This makes D(M) an object of C(G, H, α).

Clearly, D is a functor. If M and N are objects of Z(G, α)k(G/H), then

2 D M ⊗k(G/H) N = M ⊗k(G/H) N p = M ⊗k(G/H) N p =

= (Mp) ⊗k(G/H) (N p) = D(M) ⊗ D(N), whence D is a tensor functor. The functor E requires a lengthier setup. For V ∈ C(G, H, α), let VG be the vector space of set-theoretic functions a : G → V. Deﬁne G-grading on VG by

|a| = f ∈ G ⇐⇒ |a(x)| = x−1 f x ∀x ∈ G. (3.10)

Deﬁne an α-projective G-action on VG as follows. For a homogeneous a ∈ VG of degree |a| = f , deﬁne g.a : G → V by

− −1 − (g.a) (x) = α x 1, g f a g 1 x .

The action property follows from equation (1.9):

1 (uv) .a (x) = a (uv)−1 x . αx−1, uv| f

Observe that

−1 −1 (g.a) (x) = a g x = g |a(x)| g 62

∈ G ∈ G G Z So g.a Vg f g−1 for a V f . All this makes V an object of (G, α). Now consider a subspace of VG given by −1 −1 −1 E(V) = a : G → V a(xh) = α(h , x | f )h .a(x), h ∈ H, x ∈ G, |a| = f , (3.11)

for V ∈ C(G, H, α). Since the defining condition is homogeneous, E(V) is a graded subspace of VG. Moreover, E(V) is invariant under the α-projective G-action on VG: 1 α(h−1, x−1g| f ) (g.a)(xh) = a(g−1 xh) = h−1. a g−1 x = α(h−1 x−1, g| f ) α(h−1 x−1, g| f ) −1 −1 −1 α(h , x |g f g ) − − − − − − = h 1. a g 1 x = α h 1, x 1 g f g 1 h 1. (g.a) (x) . α(x−1, g| f ) Thus E(V) is also an object of Z(G, α). To make E(V) into a right k(G/H)-module, we must make sense of a morphism ν : E(V) ⊗ k(G/H) → E(V) in Z(G, α) satisfying the usual module multiplication properties. To do this, we first identify k(G/H) with E(k). We can then define ν : E(V) ⊗ E(k) → E(V) by ν(a ⊗ b) = ab, where (ab)(x) = a(x) · b(x), rescaling the vector a(x) ∈ V by the scalar b(x) ∈ k. Therefore E is a functor

C (G, H, α) → Z(G, α)k(G/H). For V, W ∈ C(G, H, α), the universal property of the tensor product gives an isomorphism

E(V) ⊗k(G/H) E(W) = E(V) ⊗E(k) E(W) ' E(V⊗W),

which shows that E is a monoidal functor. We now proceed to deﬁning the adjunction isomorphisms. Given V ∈ C(G, H, α),

deﬁne a map φV : E(V)p → V by φV (ap) = a(e), where e ∈ G is the identity element. Observe that, since e ∈ H, we have p(e) = 1, and so a(e) = a(e)p(e). We must show that

φV is a morphism in the category C(G, H, α), i.e., φV is a k-linear map that preserves the

G-grading and H-action. Clearly, φV is k-linear. Suppose that a ∈ E(V) satisﬁes |a| = g.

Then a(e) ∈ Ve−1ge = Vg, and hence |a(e)| = g. Since a(e) = φV (ap), it follows that φV preserves the G-grading. Let h ∈ H, and consider

−1 −1 −1 −1 −1 φV (h.(ap)) = (h. (ap)) (e) = (ap) h e = (ap) h = a h p h = a h = 63

= h. (a(e)) = h. (φV (ap)) .

Therefore φV preserves the H-action, and hence φV is indeed a morphism in the category

C(G, H, α). It remains to show that φV is bijective. We claim that the map ηV : V → E(V)p deﬁned by

ηV (v) = v˜, where   −1  x .v, x ∈ H v˜(x) =  ,   0, otherwise

is the inverse of φV . Observe thatv ˜ = vp˜ . To see that ηV is indeed the inverse of φV , let ap ∈ E(V)p for some a ∈ E(V). Then for x ∈ G, we have    a(x), x ∈ H (ap) (x) =  .   0, otherwise

Then ap 7→ a(e) 7→ ag(e), where   −1  x . (a(e)) , x ∈ H ag(e)(x) =  .   0, otherwise By deﬁnition of the action, this is equal to    a(x), x ∈ H  ,   0, otherwise which is clearly equal to (ap)(x). This shows that ηV ◦ φV = IdE(V)p. On the other hand, given v ∈ V, we obtain v 7→ v˜ 7→ v˜(e).

−1 By deﬁnition,v ˜(e) = e .v = e.v = v, and so φV ◦ ηV = IdV . Therefore φV is indeed

−1 invertible, and ηV = φV , as claimed. 64

Now let M ∈ Z(G, α)k(G/H). Deﬁne a map ψM : E(Mp) → M by

1 X ψ (a) = α g, g−1 |a| g.(a(g)). M |H| g∈G

We show that ψM is an isomorphism by exhibiting an inverse, and showing that that

inverse is a morphism in Z(G, α)k(G/H). Deﬁne a map ξM : M → E(Mp) by

−1 ξM (m) (x) = x .m p, x ∈ G. We ﬁrst claim that ξM is a morphism in Z(G, α)k(G/H), i.e., that ξM is a G-graded k-linear map that preserves the given G-action and is a homomorphism of right k(G/H)-modules. k-linearity of ξM is clear. Let m ∈ M. For x ∈ G, we have

−1 −1 −1 −1 |ξM (m)| = x .m p = x .m · |p| = x |m| x · e = x |m| x.

According to the G-grading on E(−), this means that |ξM(m)| = |m|, i.e., ξM preserves the G-grading.

Next, we show that ξM preserves the G-action. Suppose that |m| = f for some f ∈ G. For g, x ∈ G we have

1 (ξ (g.m)) (x) = x−1.(g.m) p = x−1g .m p = M αx−1, g| f

1 −1 1 = g−1 x .m p = ξ (m) g−1 x = g.(ξ (m)) (x), αx−1, g| f αx−1, g| f M M

i.e., ξM preserves the G-action.

Next, we show that ξM is a homomorphism of right k(G/H)-modules. Let m be as above, and let s ∈ k(G/H). Then for x ∈ G,

−1 −1 −1 −1 ξM(m · s)(x) = x .(m · s) p = x .m p x .s (H) = x .m p · s(xH).

On the other hand,

−1 (ξM(m) · s) (x) = ξM(m)(x) · s(xH) = x .m p · s(xH), 65 so that ξM(m · s) = ξM(m) · s. Thus ξM is a morphism in Z(G, α)k(G/H).

Finally, we claim that ξM is the inverse of ψM. For m ∈ M f , consider

1 X 1 X (ψ ◦ ξ )(m) = α(g, g−1| f )g.(ξ (m)(g)) = α(g, g−1| f )g. g−1.(m)p = M M |H| M |H| g∈G g∈G

1 X α(g, g−1| f ) 1 X 1 X = gg−1 .m g.p = m · p g−1H = m = |H| α(g, g−1| f ) |H| |H| g∈G g∈G h∈H 1 = |H| · m = m, |H| and therefore ψM ◦ ξM = IdM.

α(g, g−1| f ) = α(x−1g, g−1| f ). α(x−1, g|g−1 f g)

Also, for any g ∈ G, we have a(g) = a(g) · p, so that

−1 −1 g−1 x .(a(g)) p = g−1 x .(a(g) · p) p =

− − − −1 1 −1 1 −1 1 = g x .(a(g)) g x .p p = δg−1 x,H g x .(a(g)) p.

We can therefore rewrite (3.12) as

1 X −1 α(x−1g, g−1| f ) g−1 x .(a(g)) p. (3.13) |H| g∈xH 66

Now set h = g−1 x ∈ H, and note that

− − − −1 − − − α x 1g, g 1 f g 1 x .(a(g)) = α h 1, g 1 f h 1.(a(g)).

Observe that we have a(x) = a gg−1 x = a(gh), so by H-equivariance of a,

− − − a(x) = a(gh) = α h 1, g 1 f h 1.(a (g)).

Using this, we may rewrite (3.13) as

1 X a(x) · p. (3.14) |H| g∈xH Finally, (3.14) coincides with

1 X 1 1 a(x) = |xH| · a(x) = |H| · a(x) = a(x), |H| |H| |H| g∈xH

as desired. This shows that ξM ◦ ψM = IdE(Mp). Therefore ψM is indeed an isomorphism

−1 with inverse ξM = ψM , as claimed. Therefore the categories Z(G, α)k(G/H) and C(G, H, α) are tensor equivalent. At the very end of the proof, we address the locality statement. For an object V ∈ Z(G, α), we denote by supp(V) the support of V: the set

supp(V) = g ∈ G Vg , {0} .

Let M be a right k(G/H)-module, and p be as above. If m ∈ Mg for some g ∈ G, then by (1.1) and (1.11), M is local if and only if we have

mp = m(g.p) and mp = mp2 = m(p · (g.p)).

By deﬁnition of the action, g.p = p g−1H , and hence g.p , 0 if and only if g ∈ H. It follows that supp(Mp) ⊆ H. Moreover, if mp , 0, then necessarily p · (g.p) , 0, which 67 can happen if and only if g.p = p, i.e., g ∈ stabG(p) = H. As we have already shown, D(M) = Mp is a G-graded vector space with H-action. Combining these two facts, we have that M is local if and only if D(M) = Mp is, in fact, H-graded with H-action; i.e., M is local if and only if D(M) is an object of the category Z(H, α|H) of H-graded k-vector

loc spaces with H-action. This shows that Z(G, α)k(G/H) is tensor equivalent to Z(H, α|H). It is straightforward to see that this tensor equivalence is braided, as claimed, which

completes the proof.

We mention an immediate consequence of theorem 3.2.2.2.

Corollary 3.2.2.3. For a ﬁnite group G and a subgroup H ≤ G, the function algebra

A = k(G/H) in the category Z(G, α) is Lagrangian if and only if H = {e}.

loc Proof. An algebra A in Z(G, α) is Lagrangian if and only if Z(G, α)A ' k-Vect. By

loc theorem 3.2.2.2, Z(G, α)k(G/H) 'Z(H, α|H), and clearly Z(H, α|H) ' k-Vect if and only if

H = {e}.

Remark 3.2.2.4. Theorem 3.2.2.2 in combination with lemma 2.2.1.1 gives an

interpretation of the transfer, deﬁned in [43]. The transfer turns an algebra from Z(H, α|H)

into an algebra from Z(G, α). Indeed, by theorem 3.2.2.2, an algebra from Z(H, α|H) is an

loc algebra in Z(G, α)k(G/H), which by lemma 2.2.1.1 gives an algebra in Z(G, α).

Corollary 3.2.2.5 ([12], Corollary 3.9). For a simple separable algebra A in Z(G, α) there is a subgroup H ≤ G such that A is the transfer of a simple separable algebra B in

Z(H, α|H) with Be = k.

Proof. The subalgebra Ae is an indecomposable commutative G-algebra. By lemma 3.2.2.1, it is isomorphic to k(X) for some transitive G-set X. By lemma 2.2.1.1, A is a Z loc commutative algebra in (G, α)Ae . Thus, by theorem 3.2.2.2, A is the transfer of the etale

algebra B = Ap from Z(H, α|H). Here, p is the minimal idempotent of Ae, corresponding 68 to an element of X, with the stabilizer H = stabG(p). Finally, Be = Ae p = k by minimality of p.

3.2.3 Commutative Separable Algebras Trivial in Trivial Degree and Their Local Modules

Here we describe simple commutative separable algebras B in Z(H, β) with Be = k.

Proposition 3.2.3.1 ([12], Lemma 3.10 and Proposition 3.11). Let B be a separable algebra in Z(H, α|H) such that Be = k. Then B is a twisted group algebra of the form

2 ∗ k F, γ , where F = supp(B) E H and γ ∈ C (F, k ) is a 2-cochain with ∂(γ) = α|F. H-action on kF, γ is given by

h.e f = εh( f )eh f h−1 , where ε : H × F → k∗ satisﬁes each of the following identities:

−1 εgh( f ) = εg(h f h )εh( f )α(g, h| f ), g, h ∈ H, f ∈ F (3.15)

−1 −1 γ( f, g)εh( f g) = α( f, g|h)εh( f )εh(g)γ(h f h , hgh ), h ∈ H, f, g ∈ F (3.16)

If, in addition, B is commutative, then ε must also satisfy

−1 γ( f, g) = ε f (g)γ( f g f , f ), f, g ∈ F. (3.17)

Moreover,

dimk(Bh) ≤ 1 ∀h ∈ H.

Proof. An H-graded algebra B such that Be = k is separable if and only if the multiplication map µ deﬁnes a nondegenerate pairing m : Bg ⊗ Bg−1 → Be = k. Thus,

α-associativity of multiplication implies that, for any a, c ∈ Bg and b ∈ Bg−1 , we have

α g, g−1, g m(a, b)c = am(b, c). 69

Given nonzero a and c, by choosing b such that m(a, b), m(b, c) , 0, we obtain that a and c are proportional. It follows from the nondegeneracy of m : Bg ⊗ Bg−1 → Be = k that a generator of a nonzero B f is invertible, and hence F is closed under inversion. Thus, for nonzero components B f and Bg, the product B f Bg ⊆ B f g is also nonzero, which shows that F is closed under multiplication. Therefore F ≤ H. Suppose that f ∈ F and h ∈ H.

Consider Bh f h−1 . Since B is H-graded, H acts on the set of graded components of B via

0 h.(Bh0 ) = Bhh0h−1 , h, h ∈ H.

Hence Bh f h−1 = h. B f . Since f ∈ F = supp(B), we must have B f , {0}. Since h is invertible, we must have that h. B f = Bh f h−1 , {0} as well. Therefore F is closed under

conjugation, so that F E H, as claimed. Since the action of H is by assumption α-projective, we must have that (gh) .e f = α(g, h| f )g. h.e f . By deﬁnition of the action, this is equal to εgh( f )egh f h−1g−1 . We therefore obtain

εgh( f )egh f h−1g−1 = α(g, h| f )g. h.e f = α(g, h| f )g. εh( f )eh f h−1 = α(g, h| f )εh( f )g.eh f h−1

−1 = α(g, h| f )εh( f )εg h f h egh f h−1g−1 , which proves the ﬁrst identity. Multiplicativity of the action amounts to the equality between

h(e f eg) = γ( f, g)εh( f g)eh f gh−1 and

−1 −1 α( f, g|h)h(e f )h(eg) = α( f, g|h)εh( f )εh(g)γ(h f h , hgh )eh f gh−1 , which gives the second identity. Finally, commutativity implies that e f eg = γ( f, g)e f g is equal to

−1 f (eg)e f = ε f (g)e f g f −1 e f = ε f (g)γ( f g f , f )e f g. (3.18)

Denote by k[F, γ, ε] an etale algebra in Z(H, α|H), as deﬁned in proposition 3.2.3.1.

Lemma 3.2.3.2 ([12], Lemma 3.12). Two algebras kF, γ, ε and kF0, γ0, ε0 in the

∗ category Z(H, α|H) are isomorphic if and only if there is a cochain c : F → k such that

0 −1 0 c( f g)γ( f, g) = γ ( f, g)c( f )c(g), εh( f )c(h f h ) = c( f )ε h(g).

Proof. Indeed, isomorphic algebras in Z(H, α|H) have to have the same supports. Thus F = F0. Since the components of both k[F, γ, ε] and k[F0, γ0, ε0] are all one-dimensional,

0 0 0 ∗ an isomorphism k[F, γ, ε] → k[F , γ , ε ] has the form e f 7→ c( f )e f for some c( f ) ∈ k . Finally, multiplicativity of this mapping is equivalent to the ﬁrst condition, while

H-equivariance is equivalent to the second. Note that ε can be considered as an element of C1 H, C1(F, k∗) , while γ lies naturally in C2(F, k∗) = C0 H, C2(F, k∗) . Thus, in the terminology of the Appendix of [8], (ε, γ) is a

2-cochain of Ce∗(H, F, k∗). The equations (3.15),(3.16), together with the condition

∂γ = α|F, are equivalent to the coboundary condition ∂(ε, γ) = (α2, α1, α0) = τ(α) in Ce∗(H, F, k∗). The equations (3.18) say that (ε, γ) = ∂(c)(ε0, γ0) for c ∈ C1(F, k∗) = Ce1(H, F, k∗). Before we describe local modules over the algebras, we need to explain how the cochains γ and ε associated to them allow us to reduce the cocycle α ∈ Z3(H, k∗) to α ∈ Z3(H/F, k∗). It will be shown, during the proof of theorem 3.2.3.3, that α(x, y, z) deﬁned by

α(s(x), s(y), s(y)−1 s(x)−1 s(xyz))γ(τ(y, z), τ(x, yz)))γ(τ(y, z)τ(x, yz), τ0(x, y, z))×

0 −1 0 × εs(xyz)−1 s(x)s(y)(τ(x, y))γ(τ (x, y, z) , τ (x, y, z)) (3.19)

is a 3-cocycle of H/F. Here s : H/F → H is a section of the canonical projection H → H/F, τ(y, z) = s(z)−1 s(y)−1 s(yz) ∈ F and

τ0(x, y, z) = s(xyz)−1 s(x)s(y)τ(x, y)−1 s(y)−1 s(x)−1 s(xyz). 71

Our strategy in the following proof will diﬀer from that which we used in the proof of theorem 3.2.2.2 above (q.v.). In what follows, we will compute local modules over etale algebras ﬁrst, and then use the result to single out those etale algebras which are Lagrangian. We then compute all modules for these Lagrangian algebras only.

Theorem 3.2.3.3 ([8], Theorem 3.5.4; [12], Theorem 3.13). Let kF, γ, ε be the algebra

loc in Z(H, α|H), as deﬁned in proposition 3.2.3.1. Then the category Z(H, α|H)k[F,γ,ε] of local right k[F, γ, ε]-modules in Z(H, α|H), is equivalent, as a ribbon category, to Z(H/F, α), where α is as deﬁned above. M Proof. The structure of a right k[F, γ, ε]-module on an object M = Mh of Z(H, α|H) h∈H amounts to a collection of isomorphisms e f : Mh → Mh f (right multiplication by e f ∈ k[F, γ, ε]) such that

0 −1 0 ee = I, e f e f 0 = γ( f, f )e f 0 f , he f h = εh( f )eh f h−1 , f, f ∈ F, h ∈ H.

Here h : Mh0 → Mhh0h−1 is the α-projective H-action on M. The k[F, γ, ε]-module M is

−1 local iﬀ e f = εh( f )h f h eh f h−1 on Mh.

Indeed, the double braiding in Z(H, α|H) transforms an element m ⊗ e f ∈ M ⊗ A

(with m ∈ Mh) as follows:

−1 m ⊗ e f 7→ h(e f ) ⊗ m = εh( f )eh f h−1 ⊗ m 7→ εh( f )h f h (m) ⊗ eh f h−1 .

An equivalent way of expressing the locality condition is the following:

−1 −1 −1 −1 −1 −1 f = εh(h f h) γ(h f h, f )γ( f, f ) eh−1 f h f −1 = εh( f )e[h−1, f ].

Now let s : H/F → H be a section of the quotient map H → H/F. For a

k[F, γ, ε]-module M deﬁne an H/F-graded vector space V by Vx = Ms(x), where x ∈ H/F. For local M the vector space V can be equipped with a projective H/F-action: for

y ∈ H/F deﬁne y : Vx → Vyxy−1 as the composition

s(y) e f (x,y) Vx = Ms(x) / Ms(y)s(x)s(y)−1 / Ms(yxy−1) = Vyxy−1 , 72 where f (x, y) = s(y)s(x)−1 s(y)−1 s(yxy−1) = s(y) s(x)−1 s(y x) ∈ F.

To compute the projective multiplier one would need to compare zy on Vx with the composition of y and z. This can be done with the help of the following diagram:

y z Vx / Vy x / Vzy x

Ms(y x) 9 e f (x,y) s(z)

& Ms(y) s(x) Ms(z) s(y x) s(y) : s(z) es(z) f (x,y) 9 e f (y x,y)

s(z)s(x) % eg(x,y,z) % Ms(x) / Ms(z)s(y) s(x) / Ms(zy x) G W 8 e −1 σ(z,y) [s(zy) s(x) ,σ(z,y)] s(zy) f (x,zy) - Ms(zy) s(x)

zy Vx / Vzy x

Here, σ(z, y) = s(z)s(y)s(zy)−1 ∈ F and

g(x, y, z) = s(z) f (x, y) f (y x, z) = s(z)s(y) s(x)−1 s(zy x)

so that −1 [s(zy) s(x) , σ(z, y)]s(z)s(y) s(x)−1 s(zy x)

coincides with s(zy) s(x)−1 s(zy x) = f (x, zy).

The cells of the diagram commute up to multiplication by a scalar (except two top and one bottom cells, which commute on the nose). Carefully gathering the scalars one can write down the multiplier for the projective H/F-action on V. Fortunately, we do not have to do it. In view of the proposition proved in the appendix of [8], it is enough to know that such a multiplier exists. 73

The H/F-graded vector space V ⊗ U, corresponding to the tensor product

M ⊗k[F,γ,ε] N of local B = k[F, γ, ε]-modules, can be identiﬁed with the graded tensor product of V and U. Indeed, the composition

L Ms(y) ⊗ Ns(z) (M ⊗B N)s(x) yz=x O

I⊗eτ(y,z) pr L M f ⊗ Ng (M ⊗ N)s(x) f g=s(x) M deﬁnes an isomorphism Vy ⊗ Uz → (V ⊗ U)x. Here, as before, yz=x τ(y, z) = s(z)−1 s(y)−1 s(yz) ∈ F. Again, we will use a diagrammatic language to prove the compatibility (up to a scalar) of the H/F-action on V ⊗ U with the H/F-actions on V and U:

y (V ⊗ U)x / (V ⊗ U)yxy−1

s(y) (M ⊗B N)s(x) / (M ⊗B N)s(y)s(x)s(y)−1 e / (M ⊗B N)s(yxy−1) O O f (x,y) O pr pr pr s(y) (M ⊗ N)s(x) / (M ⊗ N)s(y)s(x)s(y)−1 (M ⊗ N)s(yxy−1)

L s(y)⊗s(y) L e f (v,y)⊗eh(v,y,g) L M f ⊗ Ng / Ms(y) f s(y)−1 ⊗ Ns(y)gs(y)−1 / M f 0 ⊗ Ng0 f g=s(x) f g=s(x) f 0g0=s(yxy−1) O O ⊗ ⊗ O Id⊗eτ(v,u) Id es(y)τ(v,u)s(y)−1 Id eτ(yvy−1,yuy−1)

L s(y)⊗s(y) L L Ms(v) ⊗ Ns(u) / Ms(y)s(v)s(y)−1 ⊗ Ns(y)s(u)s(y)−1 Ms(yvy−1) ⊗ Ns(yuy−1) vu=x vu=x vu=x O −1 ⊗ −1 e f (v,y) e f (u,y) L L L Vv ⊗ Uu Ms(yvy−1) ⊗ Ns(yuy−1) Vyvy−1 ⊗ Uyuy−1 vu=x vu=x vu=x

y⊗y L) Vyvy−1 ⊗ Uyuy−1 vu=x 74

Here h(v, y, g) = s(y)gs(y)−1 f (v, y)−1 s(y)g−1 s(y)−1. Again the cells of the diagram commute up to scalars (one for each v and u), resulting in a overall factor, rescaling y ⊗ y into y on V ⊗ U. Note that the six vertex cell in the right half of the diagram commutes by the following property of the projection map: for any u ∈ F the diagram

(M ⊗B N) pr 7 g pr

(M ⊗ N) (M ⊗ N)

L L M f ⊗ Ng / M f 0 ⊗ Ng0 ⊕eu⊗e −1 −1 f,g g u g f 0,g0

commutes up to multiplication by scalars (one for each f and g). So far their actual form was not important, but it will become crucial in what we are going to do later. To calculate this factor we start with the identity

meu ⊗ n = εh−1 (u)(m ⊗ neh−1uh),

which follows from the deﬁnition of the tensor product over B. Multiplying this identity

with eh−1u−1h (from the right) we get

−1 −1 −1 meu ⊗ neh−1u−1h = εh−1 (u)γ(h uh, h u h)(m ⊗ n). 75

Tha last step we need to make is to calculate the associator on V ⊗ (U ⊗ W) and to

prove that it is scalar on Vx ⊗ (Uy ⊗ Wz). Once again we apply diagrammatic technique:

α(x,y,z) Vx ⊗ (Uy ⊗ Wz) / (Vx ⊗ Uy) ⊗ Wz

Ms(x) ⊗ (Ns(y) ⊗ Ls(z)) (Ms(x) ⊗ Ns(y)) ⊗ Ls(z)

Id⊗Id⊗eτ(y,z) Id⊗eτ(x,y)⊗Id Ms(x) ⊗ (Ns(y) ⊗ Ls(y)−1 s(yz)) (Ms(x) ⊗ Ns(x)−1 s(xy)) ⊗ Ls(z)

Id⊗Id⊗eτ(x,yz) Id⊗Id⊗eτ(xy,z) Ms(x) ⊗ (Ns(y) ⊗ Ls(y)−1 s(x)−1 s(xyz)) (Ms(x) ⊗ Ns(x)−1 s(xy)) ⊗ Ls(xy)−1 s(xyz) _ 5 _ a ∗ ) (Ms(x) ⊗ Ns(y)) ⊗ Ls(y)−1 s(x)−1 s(xyz) _

M ⊗ (N ⊗ L) / (M ⊗ N) ⊗ L (M ⊗ N) ⊗ L αM,N,L pr pr pr u M ⊗ (N ⊗ L) / (M ⊗ N) ⊗ L B B αM,N,L B B

Here a stands for the multiplication by α(s(x), s(y), s(y)−1 s(x)−1 s(xyz)),

0 −1 0 ∗ = εs(xyz)−1 s(x)s(y)(τ(x, y))γ(τ (x, y, z) , τ (x, y, z))(Id ⊗ eτ(x,y) ⊗ eτ0(x,y,z)), and again τ0(x, y, z) = s(xyz)−1 s(x)s(y)τ(x, y)−1 s(y)−1 s(x)−1 s(xyz).

Now composing the arrows of the top cell and comparing the coeﬃcients gives the formula (3.19). Finally, by the proposition from the appendix of [8], the category

loc Z(H, α|H)k[F,γ,ε] is ribbon equivalent to Z(H/F, α).

We close this section by mentioning an immediate consequence of theorem 3.2.3.3.

Corollary 3.2.3.4 ([12], Corollary 3.14). An etale algebra B = kF, γ, ε in the category

Z(H, α|H) is Lagrangian if and only if F = H. 76 Proof. By theorem 3.2.3.3, an etale algebra B = k F, γ, ε in Z(H, α|H) is Lagrangian if and only if Z(H/F, α) ' k-Vect, i.e., if and only if the quotient group H/F is trivial,

which happens if and only if F = H.

3.2.4 Etale Algebras and Their Local Modules

In this section, we combine the previous results on commutative separable algebras in group-theoretical modular categories and on their local modules. Deﬁne

A(H, F, γ, ε) = EkF, γ, ε, where E is the functor from the proof of theorem 3.2.2.2 above.

Theorem 3.2.4.1 ([12], Theorem 3.15). Etale algebras in Z(G, α) correspond to

quadruples (H, F, γ, ε), where H ≤ G is a subgroup, F E H is a normal subgroup, 2 ∗ ∗ γ ∈ C (F, k ) is a coboundary ∂γ = α|F and ε : H × F → k satisﬁes the conditions (3.15), (3.16), and (3.17).

Proof. Follows from corollary 3.2.2.5 and lemma 2.2.1.1.

loc Theorem 3.2.4.2 ([8], Theorem 3.6.3; [12], Theorem 3.16). The category Z(G, α)A(H,F,γ,ε) of local right A(H, F, γ, ε)-modules in Z(G, α), is equivalent, as a ribbon category, to Z(H/F, α).

Proof. Follows from lemma 2.2.1.1 and theorems 3.2.2.2 and 3.2.3.3.

Remark 3.2.4.3. When F = H, the function ε is completely determined by γ. Indeed, by the commutativity condition (3.17), we have γ( f, g) ε (g) = , f, g ∈ H. f γ f g f −1, f Assume now that F = H. Write L(H, γ) in place of A(H, H, γ, ε). Theorems 3.2.4.1 and 3.2.4.2 allow us to describe Lagrangian algebras in group-theoretical modular categories in purely group-theoretical terms. This is the next major result: 77

Corollary 3.2.4.4 ([12], Corollary 3.17). Lagrangian algebras L ∈ Z(G, α) correspond to

2 ∗ pairs (H, γ), where H ≤ G is a subgroup and γ ∈ C (H, k ) is a coboundary ∂γ = α|H.

Proof. Follows from corollary 3.2.3.4 and theorems 3.2.4.1 and 3.2.4.2.

Remark 3.2.4.5. A Lagrangian algebra L = L(H, γ) is completely characterized by the following conditions: the trivial-degree component Le is isomorphic to the function algebra k(G/H), and the image D(L) under the functor D from the proof of theorem 3.2.2.2 is isomorphic to the twisted group algebra kH, γ ∈ C(G, H, α).

3.3 The Full Center of an Algebra Redux

3.3.1 The Full Center of the Twisted Group Algebra

Theorem 3.3.1.1 ([12], Theorem 4.2). Let k[H, γ] be the twisted group algebra, viewed as an algebra in V(G, α). Then the full center Z(k[H, γ]) coincides with L(H, γ) = EkH, γ.

Proof. It suffices to construct a homomorphism of algebras ζ : L(H, γ) → k[H, γ] in V(G, α) fitting in the diagram (1.2). Indeed, such a homomorphism induces a homomorphism of algebras ζ˜ : L(H, γ) → Z(k[H, γ]) in Z(G, α). Since L(H, γ) is a separable indecomposable algebra in Z(G, α), ζ˜ is a monomorphism by lemma 2.1.2.1. Finally, the dimension of Z(G, α) is |G|, which coincides with the dimension of the full center Z(k[H, γ]). Let us define a map ζ : EkH, γ → kH, γ by ζ(a) = a(e). Observe that this definition, effectively, implies that ζ(a) = 0 if |a| ∈ G \ H. We claim that ζ is a homomorphism of algebras in the category V(G, α). As ζ is an evaluation map, it is automatically additive and multiplicative. It remains to check that ζ is G-graded. Let a ∈ EkH, γ such that |a| = f ∈ G. Recall from the proof of theorem 3.2.2.2 that

|a| = f ∈ G ⇐⇒ |a(x)| = x−1 f x ∀x ∈ G, 78 or equivalently, |a| = x |a(x)| x−1 ∀x ∈ G.

From this, we see that

|ζ(a)| = |a(e)| = e−1|a|e = e−1 f e = f = |a|,

and therefore ζ does indeed preserve the G-grading, so that ζ is a morphism in V(G, α), as claimed. Now we consider the diagram (1.2). Here, A = kH, γ and Z = L(H, γ), so that the diagram becomes

Id⊗ζ ⊗2 k H, γ ⊗ L(H, γ) / k H, γ (3.20) µ

% c k H, γ 9 µ ⊗2 L(H, γ) ⊗ k H, γ / k H, γ ζ⊗Id Let a ∈ L(H, γ), and let eh ∈ k H, γ for some h ∈ H. Going the short way, we obtain

eh ⊗ a 7→ eh ⊗ a(e) 7→ eha(e). (3.21)

Going the long way, we obtain

eh ⊗ a 7→ (h.a) ⊗ eh 7→ ((h.a) (e)) ⊗ eh 7→ ((h.a) (e)) eh. (3.22)

We can use H-equivariance of a to rewrite the rightmost side of (3.22) as

−1 ((h.a) (e)) eh = a h eh = h. (a(e)) eh (3.23)

The rightmost sides of (3.21) and (3.23) coincide because the H-action on kH, γ is inner in the sense that, for y ∈ kH, γ and h ∈ H, we have

−1 h.y = ehy (eh) . (3.24) 79

It therefore follows that ζ makes the diagram (3.20) commute. This completes the proof.

Remark 3.3.1.2. It follows from theorem 3.2.4.4 that Lagrangian algebras in the Deligne

−1 −1 product Z(G, α) Z G, α 'Z G × G, α × α correspond to pairs (U, γ), where U ≤ G × G is a subgroup and γ ∈ C2(U, k∗) is a coboundary ∂γ = α × α−1 . This U coincides with the parametrization of module categories obtained in [38], which illustrates the fact39 that the full center deﬁnes a bijection between equivalence classes of indecomposable module categories over Z(G, α) and Lagrangian algebras in

−1 Z(G, α) Z G, α .

3.4 Characters of Lagrangian Algebras

3.4.1 Irreducible Characters

For an object Z ∈ Z(G, α), deﬁne its character as the following function on pairs of commuting elements of G:

χZ( f, g) = TrZ f (g).

Lemma 3.4.1.1 ([12], Lemma 5.1). Let Z ∈ Z(G, α). Then the character χZ satisﬁes

α(xgx−1, x| f ) χ x f x−1, xgx−1 = · χ ( f, g) (3.25) Z α(x, g| f ) Z

Proof. Let ρ : G → Aut(Z) be the representation homomorphism corresponding to the action of G on the graded vector space Z. For f, g, x ∈ G, we have, by deﬁnition,

χ ( f, g) = Tr (ρ(g)) = Tr ρ(x)ρ(g)ρ(x)−1 . Z Z f Zx f x−1

Note that by (1.6), we can write

α(xgx−1, x| f ) ρ(x)ρ(g)ρ(x)−1 = ρ xgx−1 . α(x, g| f )

39 As formulated in [8, § 2.5]. 80

Indeed, 1 ρ(x)ρ(g) = ρ(xg). α(x, g| f ) Similarly, 1 ρ xgx−1 ρ(x) = ρ(xg). α(xgx−1, x| f ) Combining these, we see that

−1 −1 α(xgx , x| f ) −1 TrZ ρ(x)ρ(g)ρ(x) = TrZ ρ xgx = x f x−1 α(x, g| f ) x f x−1

α(xgx−1, x| f ) = χ x f x−1, xgx−1 . α(x, g| f ) Z

The result follows.

Remark 3.4.1.2. Equation (3.25) implies that a character of Z(G, α) can be nonzero only on those commuting pairs ( f, g) ∈ G × G for which

α(x, f, g)α(g, x, f )α( f, g, x) = 1 α(x, g, f )α( f, x, g)α(g, f, x) for all x ∈ CG( f ) ∩ CG(g).

By a character of Z(G, α), we mean a function on pairs of commuting elements of G satisfying the projective class function property (3.25). For characters χ and χ0, deﬁne the product χχ0 by

0 X 0 χχ ( f, g) = α(g| f1, f2)χ( f1, g)χ ( f2, g). (3.26)

f1 f2= f fig=g fi

Lemma 3.4.1.3 ([12], Lemma 5.3). Let Z, W ∈ Z(G, α), and let χZ, χW denote their

respective characters. Then χZχW = χZ⊗W .

Proof. We have X χ ⊗ ( f, g) = Tr ⊗ (g) = Tr ⊗ (g). Z W (Z W) f Z f1 W f2 f1 f2= f 81

If fi < CG(g), then the corresponding summand contributes zero to the trace. Otherwise,

X X Tr ⊗ (g) = α(g| f , f )Tr (g)Tr (g) = Z f1 W f2 1 2 Z f1 W f2 f1 f2= f f1 f2= f X = α(g| f1, f2)χZ( f, g)χW ( f, g) = χZχW ,

f1 f2= f fig=g fi as desired.

Remark 3.4.1.4. The character of the dual object (the dual character) has the form

1 −1 −1 χ ∗ ( f, g) = χ f , g . X α(g| f, f −1)α(g−1, g| f −1) X

As in the ordinary character theory, the space of characters of Z(G, α) comes equipped with a scalar product (see [1]):

1 X 1 χ, χ0 = χ f, g−1 χ0( f, g). |G| α(g−1, g| f ) f,g∈G, f g=g f

The scalar product calculates dimensions of corresponding Hom-spaces in Z(G, α), as the next lemma shows.

Lemma 3.4.1.5 ([12], Lemma 5.5). Let Z, W ∈ Z(G, α). Then

(χZ, χW ) = dim HomZ(G,α)(Z, W) .

In particular, for simple Z, W ∈ Z(G, α), the scalar product (χZ, χW ) = 1 if and only if Z ' W, and zero otherwise.

Proof. First, observe that the Hom-space HomZ(G,α)(I, W) coincides with the vector space

G of G-invariants We , so that

1 X dimHom (I, W) = dim WG = χ (e, g). Z(G,α) k e |G| W g∈G 82

∗ More generally, HomZ(G,α)(Z, W) ' HomZ(G,α)(I, Z ⊗ W), and hence

∗ 1 X dim Hom (Z, W) = dim Hom (I, Z ⊗ W) = χ ∗ (e, g) = Z(G,α) Z(G,α) |G| Z ⊗W g∈G

1 X −1 −1 = α(g| f , f )χ ∗ f , g χ ( f, g) = |G| Z W f,g∈G f g=g f 1 X α(g| f −1, f )α(g−1, g| f −1) = χ f, g−1 χ ( f, g) = |G| α(g| f, f −1) Z W f,g∈G f g=g f 1 X = α(g−1, g| f )χ f, g−1 χ ( f, g) = (χ , χ ) . |G| Z W Z W f,g∈G f g=g f

3.4.2 Characters of Lagrangian Algebras in Z(G, α)

Lemma 3.4.2.1. Let G be a group, let k be a ﬁeld, and let β ∈ Z2(G, k∗). Assume that β is a coboundary δ(x)δ(y) β(x, y) = (∂δ) (x, y) = (3.27) δ(xy) for some δ ∈ C1(G, k∗). Then the map φ : kG, β → k[G] given by

φ(ex) = δ(x)ex, x ∈ G,

is an isomorphism of algebras in k-Vect. Moreover, for M ∈ k[G]-Mod, the character χM is given by the formula

χφ−1[M](x) = δ(x)χM(x), x ∈ G. (3.28)

Proof. The map φ as deﬁned is clearly an isomorphism of k-vector spaces. It suﬃces to show that φ preserves the multiplication:

φ exey = φ β(x, y)exy = β(x, y)δ(xy)exy. 83

On the other hand,

φ(ex)φ ey = (δ(x)ex) δ(y)ey = δ(x)δ(y)exey = δ(x)δ(y)exy.

These coincide by the coboundary equation (3.27). The equation (3.28) follows as an immediate consequence.

Lemma 3.4.2.2 ([12], Lemma 5.6). For V ∈ Z(H, α|H), the character χE(V) has the form X α(y−1gy, y−1| f ) χ ( f, g) = χ y−1 f y, y−1gy , f, g ∈ G, (3.29) E(V) α(y−1, g| f ) V y∈Y where Y is a set of representatives in G of the cosets of

n − − o. y ∈ G y 1 f y, y 1gy ∈ H H ⊆ G/H.

Proof. Let V ∈ Z (H, α|H). It follows from the deﬁning condition that functions from E(V) are supported by unions of right H-cosets of G. Clearly, any function from E(V) is a unique sum of functions, each of which is supported on a single coset. For a coset yH, a function a ∈ E(V) with support supp(a) = yH is uniquely determined by its value

a(y) = v ∈ V at y. Denote such a function by ay,v. The space E(V) is spanned by the ay,vs,

where y ∈ G and v ∈ V. By (3.10), the degree of ay,v is f ∈ G if and only if v ∈ Vy−1 f y. Note

that for v ∈ Vh−1y−1 f yh, we have

−1 −1 −1 −1 α(h , y | f )α(h, h |y f y)ayh,v = ay,h.v, h ∈ H.

Indeed,

−1 −1 −1 v = ayh,v(yh) = α(h , y | f )h . ayh,v(y) ,

and thus

−1 −1 −1 −1 ay,h.v(y) = h.v = α(h , y | f )α(h, h |y f y)ayh,v(y).

We can therefore write M M D E E(V) f = ay,v v ∈ Vy−1 f y ' Vy−1 f y, (3.30) y y 84 where the sum is taken over a set of representatives of the cosets G/H.

For g ∈ CG( f ), consider the k-linear operator g : E(V) f → E(V) f . Recall from the proof of theorem 3.2.2.2 that the action of g on E(V) is given by

−1 −1 (g.a) (x) = α(x , g| f )a g x , x ∈ G, a ∈ E(V) f .

In particular, supp(g.a) = gsupp(a). Therefore g.ay,v is supported by the coset gyH and

0 hence can be written as agy,v0 for some v ∈ V. The action of g on E(V) f permutes the direct summands of (3.30) according to the left action of g on the set of cosets G/H. Thus

the trace TrE(V) f (g) has the form

X Tr (g) = Tr (g), (3.31) E(V) f hay,v|v∈Vy−1 f yi y

where the sum runs over representatives of those cosets yH for which gyH = yH. Now let y ∈ G be such that gyH = yH, or, equivalently, y−1gy ∈ H. To compute the trace

Tr (g), hay,v|v∈Vy−1 f yi

0 note that for v ∈ Vy−1 f y, g.ay,v can be written as ay,v0 for some v ∈ V. More explicitly,

0 −1 −1 −1 −1 −1 v = g.ay,v(y) = α(y , g| f )ay,v g y = α(y , g| f )ay,v yy g y =

−1 −1 X α(y gy, y | f ) −1 χE(V)( f, g) = TrE(V) (g) = TrV y gy = f α(y−1, g| f ) y−1 f y y∈Y

X α(y−1gy, y−1| f ) = χ y−1 f y, y−1gy , α(y−1, g| f ) V y∈Y which gives the desired result. 85

Lemma 3.4.2.3 ([12], Lemma 5.7). Let kH, γ be the twisted group algebra, viewed as an algebra in the category Z(H, α|H). Then the character χk[H,γ] of k H, γ is given by

γ(u, h) χ (h, u) = , hu = uh. (3.32) k[H,γ] γ(h, u) Proof. Let h ∈ H. The degree-h component k H, γ h is one-dimensional and has the form

k H, γ h = keh,

where eh is the standard basis vector of degree h in k H, γ . Now taking u ∈ H, we have

γ(u, h) u.e = e −1 . h γuhu−1, u uhu

If u ∈ CH(h), then this formula becomes

γ(u, h) u.e = e . h γ(h, u) h If u < CH(h), then u.eh < k H, γ h, so the degree-h component is not a ﬁxed point of the action and therefore contributes zero to the character. Hence by lemma 3.4.2.2, the

character χk[H,γ] has the form given by equation (3.32), as claimed.

Combining the last two results proves the main result of this section:

Theorem 3.4.2.4 ([12], Theorem 5.8). Let L = L(H, γ) be a Lagrangian algebra in the

Drinfeld center Z(G, α). Then the character χL has the form

X α(y−1gy, y−1| f )γ(y−1 f y, y−1gy) χ ( f, g) = , f, g ∈ G, L α(y−1, g| f )γ(y−1gy, y−1 f y) y∈Y

where Y is a set of representatives in G of the cosets of

n − − o. y ∈ G y 1 f y, y 1gy ∈ H H ⊆ G/H.

Proof. Follows from lemmata 3.4.2.2 and 3.4.2.3. 86

3.5 Example: Characters of Z(D3, α)

Here, we compute the characters for a relatively tractable example.

3 ∗ 3.5.1 Brief Description of H (D3, C )

We choose the the following presentation of D3:

D 3 2 −1 E D3 = r, s r = s = e; srs = r .

3 ∗ According to [5, § 6.3], the third cohomology group H (D3, C ) is cyclic of order 6. It is ∗ generated by the cohomology class of the function $ : D3 × D3 × D3 → C given by

$(sm1 rn1 , sm2 rn2 , sm3 rn3 ) = " !# 2πi 9 = exp (−1)m2+m3 n (−1)m3 n + n − [(−1)m3 n + n ] + m m m , 9 1 2 3 2 3 3 2 1 2 3

where [·]3 denotes taking the congruence class modulo 3.

2 ∗ Recall that the second cohomology group H (D3, C ) is trivial. This implies that a 3 ∗ coboundary for the restriction α|H of a 3-cocycle α ∈ Z (D3, C ) to a subgroup H ≤ D3 is

unique if it exists. It follows that Lagrangian algebras in Z(D3, $) are labeled only by conjugacy classes of subgroups on which the restriction of α is cohomologically trivial. We call such subgroups admissible subgroups. In terms of the cocycle α, there are four distinct cases, depending upon the order of

3 ∗ the cohomology class of α in H (D3, C ). In what follows, ω, ε, and η denote primitive cube, fourth, and ninth roots of unity, respectively.

3.5.2 Wherein |α| = 1

0 Taking α = $ ≡ 1 as our associator turns the twisted Drinfeld center Z(D3, α) into

the untwisted version Z(D3). The character table for Z(D3) is as follows: 87

Table 3.1: Characters of Z(D3) (e, e) (e, r)(e, s)(r, e)(r, r) r, r2 (s, e)(s, s)

χ0 1 1 1 0 0 0 0 0

χ1 1 1 −1 0 0 0 0 0

χ2 2 −1 0 0 0 0 0 0

χ3 0 0 0 1 1 1 0 0

−1 χ4 0 0 0 1 ω ω 0 0

−1 χ5 0 0 0 1 ω ω 0 0

χ6 0 0 0 0 0 0 1 1

χ7 0 0 0 0 0 0 1 −1

All subgroups are admissible in this case. Up to conjugacy, there are four subgroups:

{e}, hsi, hri, D3.

The characters of the corresponding Lagrangian algebras are as follows:

Table 3.2: Characters of Lagrangian Algebras in Z(D3) (e, e)(e, r)(e, s)(r, e)(r, r) r, r2 (s, e)(s, s)

L({e}) 4 0 0 0 0 0 0 0 L(hsi) 3 0 1 0 0 0 1 1 L(hri) 2 2 0 2 2 2 0 0

L(D3) 1 1 1 1 1 1 1 1

In view of the character table for Z(D3), the above table gives rise to the following decompositions into sums of irreducible characters: 88 Table 3.3: Decomposition into Irreducible Characters

L({e}) L(hsi) L(hri) L(D3)

χ0 + χ1 + 2χ2 χ0 + χ2 + χ6 χ0 + χ1 + 2χ3 χ0 + χ3 + χ6

3.5.3 Wherein |α| = 2

In this case, α = $3. The corresponding character table is as follows:

Table 3.4: Characters of Z(D3, α); |α| = 2 (e, e) (e, r)(e, s)(r, e)(r, r) r, r2 (s, e)(s, s)

χ0 1 1 1 0 0 0 0 0

χ1 1 1 −1 0 0 0 0 0

χ2 2 −1 0 0 0 0 0 0

χ3 0 0 0 1 1 1 0 0

−1 χ4 0 0 0 1 ω ω 0 0

−1 χ5 0 0 0 1 ω ω 0 0

χ6 0 0 0 0 0 0 1 ε

χ7 0 0 0 0 0 0 1 −ε

The admissible subgroups are {e} and hri. The characters of the corresponding Lagrangian algebras are as follows: 89

Table 3.5: Characters of Lagrangian Algebras in Z(D3, α) (e, e)(e, r)(e, s)(r, e)(r, r) r, r2 (s, e)(s, s)

L({e}) 4 0 0 0 0 0 0 0 L(hri) 2 2 0 2 1 + ω2 1 + ω 0 0

In view of the character table for Z(D3, α), the above table gives rise to the following decompositions into sums of irreducible characters:

Table 3.6: Decomposition into Irreducible Characters

L({e}) L(hri)

χ0 + χ1 + 2χ2 χ0 + χ1 + χ3 + χ5

3.5.4 Wherein |α| = 3

2 In this case, α = $ . The character table for Z(D3, α) is as follows:

Table 3.7: Characters of Z(D3, α); |α| = 3 (e, e) (e, r)(e, s)(r, e)(r, r) r, r2 (s, e)(s, s)

χ0 1 1 1 0 0 0 0 0

χ1 1 1 −1 0 0 0 0 0

χ2 2 −1 0 0 0 0 0 0

2 χ3 0 0 0 1 η η 0 0

4 8 χ4 0 0 0 1 η η 0 0

7 3 χ5 0 0 0 1 η η 0 0

χ6 0 0 0 0 0 0 1 1

χ7 0 0 0 0 0 0 1 −1 90

The admissible subgroups are {e} and hsi. The characters of the corresponding Lagrangian algebras are as follows:

Table 3.8: Characters of Lagrangian Algebras in Z(D3, α) (e, e)(e, r)(e, s)(r, e)(r, r) r, r2 (s, e)(s, s)

L({e}) 4 0 0 0 0 0 0 0 L(hsi) 3 0 1 0 0 0 1 1

In view of the character table for Z(D3, α), we have the following decomposition into irreducibles:

Table 3.9: Decomposition into Irreducible Characters

L({e}) L(hsi)

χ0 + χ1 + 2χ2 χ0 + χ2 + χ6

3.5.5 Wherein |α| = 6

In this case, α = $. The character table for Z(D3, α) is as follows: 91

Table 3.10: Characters of Z(D3, α); |α| = 6 (e, e) (e, r)(e, s)(r, e)(r, r) r, r2 (s, e)(s, s)

χ0 1 1 1 0 0 0 0 0

χ1 1 1 −1 0 0 0 0 0

χ2 2 −1 0 0 0 0 0 0

2 χ3 0 0 0 1 η η 0 0

4 8 χ4 0 0 0 1 η η 0 0

7 3 χ5 0 0 0 1 η η 0 0

χ6 0 0 0 0 0 0 1 ε

χ7 0 0 0 0 0 0 1 −ε

Only the trivial subgroup {e} is admissible in this case. Its character is identical to the untwisted case:

Table 3.11: Characters of Lagrangian Algebras in Z(D3, α) (e, e)(e, r)(e, s)(r, e)(r, r) r, r2 (s, e)(s, s)

L({e}) 4 0 0 0 0 0 0 0

Finally, the decomposition into irreducible characters is, also, identical to the untwisted case:

Table 3.12: Decomposition into Irreducible Characters

L({e})

χ0 + χ1 + 2χ2 92

3.6 Automorphisms of Lagrangian Algebras in Group-theoretical Braided Fusion Categories

Here, we make a few remarks about the automorphism groups of Lagrangian algebras in Z(G, α).

3.6.1 Lagrangian Algebras in Drinfeld Centers of Finite Groups

The following classiﬁcation of Lagrangian algebras in Z (G, α) was proved in [12] and can be found in § 3.2.4 of this document as corollary 3.2.4.4. We restate it here.

Theorem 3.6.1.1. Lagrangian algebras in the Drinfeld center Z(G, α) correspond to

2 ∗ pairs (H, γ), where H ≤ G is a subgroup and γ ∈ C (H, k ) is a coboundary ∂γ = α|H of the restriction of the associator to H.

40 2 ∗ Explicitly , let γ ∈ C (H, k ) be such that ∂γ = α|H. Denote by k[H, γ] the twisted

group algebra of H, viewed as an algebra in Z(H, α|H). Then −1 −1 −1 L(H, γ) = a : G → k[H, γ] a(xh) = α(h , x | f )h .(a(x)), h ∈ H, x ∈ G, |a| = f .

Here, |a| = f ⇐⇒ |a(x)| = x−1 f x ∀x ∈ G. The G-action on L(H, γ) is as follows: given a homogeneous a ∈ L(H, γ) of degree f ∈ G and an arbitrary g ∈ G, the function g.a : G → k[H, γ] is given by 1 (g.a)(x) = a g−1 x , x ∈ G . α(x−1, g| f ) 3.6.2 Automorphisms of Lagrangian Algebras

2 ∗ Let H ≤ G be a subgroup, and let γ ∈ C (H, k ) be such that ∂γ = α|H. The

automorphism group AutZ(G,α)(L(H, γ)) of the corresponding Lagrangian algebra ﬁts in the short exact sequence of groups

AutZ(G,α)loc (L(H, γ)) −→ AutZ(G,α)(L(H, γ)) −→ AutRep(G)(L(H, γ)e) (3.33) L(H,γ)e 40 Cf. the proof of theorem 3.2.2.2 in chapter3. 93

The trivial-degree component L(H, γ)e is the algebra k(G/H) of functions on cosets. Consequently,

AutRep(G)(L(H, γ)e) = AutRep(G)(k(G/H)) ' AutG(G/H) ' NG(H)/H . (3.34)

Z G, α loc 41 Z H, α| L H, γ 42 Since the category ( )L(H,γ)e is equivalent to ( H), and since ( ) is

k[H, γ] when viewed as an algebra in Z(H, α|H), it follows that

loc AutZ(G,α) (L(H, γ)) ' AutZ(H,α|H)(k[H, γ]) , L(H,γ)e

which is, in turn, isomorphic to the character group Hb. Thus the sequence (3.33) becomes

Hb −→ AutZ(G,α)(L(H, γ)) −→ NG(H)/H (3.35)

Recall43 the category C(G, H, α), whose objects are G-graded vector spaces with

α|H-projective H-action and associativity twisted by α, and whose morphisms are k-linear

44 maps preserving the G grading and α|H-projective H action. Recall the functor

E : C(G, H, α) → Z(G, α)k(G/H) given by

−1 −1 −1 E(V) = a : G → V a(xh) = α(h , x | f )h .(a(x)) ∀h ∈ H ∀x ∈ G; |a| = f , where the grading on E(V) is |a| = f ⇐⇒ |a(x)| = x−1 f x ∀x ∈ G, and G-action on E(V) −1 is given by (g.a)(x) = α x−1, g| f a g−1 x . It was proven45 in § 3.2.2 that the functor E is a tensor equivalence; moreover, when restricted to the full subcategory

loc Z(H, α|H) ⊆ C(G, H, α), E induces a braided equivalence Z(H, α|H) 'Z(G, α)k(G/H).

Let now ζ ∈ AutZ(G,α)(L(H, γ)), and let ζe = ζ|L(H,γ)e be the restriction to the trivial-degree component. The associated inverse image functor

41 See theorem 3.2.2.2. 42 This is an abuse of language. The image of L(H, γ) under the functor D from the proof of theorem 3.2.2.2 is isomorphic to k[H, γ]. See also remark 3.2.4.5. 43 E.g., from “Deﬁnitions” or [12, Appendix A]. 44 See the proof of theorem 3.2.2.2 in § 3.2.2. 45 See also [12, Theorem 3.7]. 94

∗ ζe : Z(G, α)k(G/H) → Z(G, α)k(G/H) gives a tensor autoequivalence of the category of ∗ modules. Moreover, since ζe preserves the locality condition (1.1), its restriction to the

loc subcategory Z(G, α)k(G/H) of local modules is a braided autoequivalence. The diagram of functors

E loc Z(H) / Z(H, α|H) / Z(G, α)k(G/H)

∗ Fζe ζe Z(H) / Z(H, α| ) / Z(G, α)loc H E k(G/H)

commutes up to a natural isomorphism. Note that ζe can be considered as an element of

NG(H)/H ≤ Out(H) using (3.34), and Fζe is a braided autoequivalence of Z(H) induced

by ζe ∈ Out(H). 95 4 Lagrangian Algebras in Pointed Categories

In this chapter, we consider the case of pointed fusion categories in detail. Lagrangian algebras in a pointed category of the form C(A, q), where A is a ﬁnite Abelian group and q is a quadratic form on A, correspond to Lagrangian subgroups L of A. These are subgroups onto which q restricts trivially46 of the largest possible order47. We derive a formula for the associator β of the group of symmetries of the Lagrangian algebra corresponding to a Lagrangian subgroup L of A.

4.1 Modular Extensions of Finite Groups

In this section, we interpret the third cohomology group H3(G, k∗) as the group of modular extensions of Rep(G), and then consider pointed modular extensions of Rep(G).

4.1.1 The Categorical Group Mex(S)

Recall48 that a braided fusion category D is nondegenerate if its symmetric center

Zsym(D) = X ∈ D cX,Y ◦ cY,X = IdY⊗X, ∀ Y ∈ D

coincides with k-Vect. Let S be a symmetric fusion category. Following [30], we say that a nondegenerate braided category D containing S as a full subcategory is a modular extension49 of S if the symmetric centralizer

CD(S) = X ∈ D cX,Y ◦ cY,X = IdY⊗X ∀Y ∈ S

coincides with S.

46 I.e., isotropic subgroups of A. 47 I.e., the order of L squares to the order of A. 48 E.g., from [11]. 49 We are abusing language here; what we call a “modular extension” would more properly be termed a “nondegenerate braided extension”. See [30, Remark 4.2]. 96

Remark 4.1.1.1. A nondegenerate braided category D containing S as a full subcategory is a modular extension if and only if dim(D) = (dim(S))2. Indeed, S is always a

subcategory of its symmetric centralizer CD(S), and dim(D) = dim(S) · dim(CD(S))

makes dim(CD(S)) = dim(S) equivalent to S = CD(S).

Deﬁne the category Mex(S) as follows. Objects of Mex(S) are modular extensions of S. Morphisms are isomorphism classes of braided equivalences preserving the subcategory S, i.e., making the diagram

C / D _ ?

commute on the nose.

Remark 4.1.1.2. If one considers isomorphisms between braided equivalences of

modular extensions as 2-cells, one naturally ends up with a 2-category Mex(S).

0 0 For D, D ∈ Mex(S), the Deligne product S S is a full subcategory of D D . The tensor product functor ⊗ : S S → S has a two-sided adjoint F : S → S S. This gives rise to a commutative algebra A = F(I) ∈ S S, where I ∈ S is the monoidal unit. This algebra is Lagrangian. Indeed, dim(A) = dim(S), and so

2 2 dim(D) = dim(S) · dim(CD(S)) = (dim(S)) = (dim(A)) .

0 0 loc Deﬁne an operation S : Mex(S) × Mex(S) → Mex(S) by D S D = (D D )A .

The category Mex(S) is monoidal with respect to S. The monoidal unit is the monoidal (or Drinfeld) center Z(S) of S with the natural embedding S → Z(S). Moreover, Mex(S) is a categorical group. Indeed, the quasi-inverse of a modular extension C with respect to S is C, the category with the inverse braiding. Denote by

Mex(S) the zeroth homotopy group π0(Mex(S)). By the definition, the first homotopy 97 group π1(Mex(S)) coincides with the group Autbr(Z(S)/S) of isomorphism classes of braided tensor autoequivalences of Z(S) fixing objects of S on the nose.

Remark 4.1.1.3. The 2-category Mex(S) from remark 4.1.1.2 is clearly a 2-categorical group. The standard invariants π0 and π1 of Mex(S) are the same as for Mex(S). It follows from the deﬁnition that π2(Mex(S)) = ker Aut⊗(IdZ(S)) → Aut⊗(IdS) .

We will be interested in the case when the base symmetric category S is the category Rep(G) of ﬁnite-dimensional representations of a ﬁnite group G. Modular extensions of Rep(G) turn out to be twisted Drinfeld centers50 of G. For a 3-cocycle α ∈ Z3(G, k∗) denote by Z(G, α) the category whose objects are G-graded k-vector spaces with α-projective G-action and whose morphisms are k-linear maps preserving the grading and α-projective action. Note that Rep(G) is a full symmetric subcategory of Z(G, α) consisting of trivially-graded objects. Moreover, dim(Z(G, α)) = |G|2 = (dim(Rep(G))2, so Z(G, α) is indeed a modular extension of Rep(G) by remark 4.1.1.1. The following result was proved in [30, Theorem 4.22]. We add a sketch of the proof for the sake of completeness.

Proposition 4.1.1.4 ([13], Proposition 2.4). The assignment H3(G, k∗) → Mex(Rep(G)) given by α 7→ Z(G, α) is an isomorphism.

Proof. The homomorphism property of the assignment is established by a braided equivalence Z(G, α) Rep(G) Z(G, β) → Z(G, αβ). Note that

Z(G, α) Z(G, β) 'Z(G × G, α × β) as braided fusion categories. Let δ : G → G × G be the diagonal embedding g 7→ (g, g). This induces a functor δ∗ : Rep(G × G) → Rep(G) making the diagram

⊗ Rep(G) Rep(G) / Rep(G) 6

δ∗ Rep(G × G)

50 See chapter3 or [12, § 2] for details. 98 commute. This functor has a two-sided adjoint δ∗. By Frobenius reciprocity, the adjoint δ∗

is just the induction with respect to δ, and so δ∗(I) = k. This implies that δ∗(k) is the function algebra51 k((G × G)/∆(G)) ∈ Z(G × G, α × β). By theorem 3.2.2.2, its category of local modules is equivalent to Z(G, αβ). Now let D be a modular extension of Rep(G). The function algebra A = k(G) ∈ Rep(G) with the regular G-action is etale in Rep(G) and in D as well.

Moreover, the group of algebra automorphisms is Autalg(A) = G. Since

dim(A) = dim(Rep(G)), A is Lagrangian. Hence the category of modules DA admits a decomposition

M g-loc DA = DA g∈G into g-local subcategories for each g ∈ G. The result is tensor equivalent52 to V(G, α) for

3 ∗ some α ∈ Z (G, k ). Thus we have D = Z(DA) 'Z(V(G, α)) 'Z(G, α) which shows

that the assignment is bijective.

The ﬁrst homotopy group π1(Mex(Rep(G))) also has a cohomological description.

2 ∗ Lemma 4.1.1.5 ([13], Lemma 2.5). π1(Mex(Rep(G))) ' H (G, k ).

Proof. By deﬁnition, π1(Mex(Rep(G))) is the group Autbr(Z(Rep(G))/Rep(G)) of braided autoequivalences of the monoidal center Z(Rep(G)) that ﬁx the full subcategory Rep(G) ⊆ Z(Rep(G)) on the nose. The monoidal center Z(Rep(G)) is equivalent, as a braided monoidal category, to the untwisted Drinfeld center Z(G). The algebra of functions k(G) ∈ Rep(G) is an etale algebra in Z(G); its category of right modules

Z(G)k(G) is equivalent to the category V(G) of G-graded vector spaces. This equivalence

gives rise to a homomorphism Autbr(Z(Rep(G))/Rep(G)) → Aut⊗(V(G)). It follows from

1 [37, Corollary 6.9] that it is an isomorphism with the group Aut⊗(V(G)) of isomorphism

51 Here, ∆(G) denotes the diagonal subgroup {(g, g)|g ∈ G} < G × G. 52 See [30, § 4.3] for details. 99 classes of tensor structures on the identity functor (the so-called soft autoequivalences).

1 2 ∗ The group Aut⊗(V(G)) is, in turn, isomorphic to H (G, k ) by [6, Proposition 2.5].

Remark 4.1.1.6. The group π2(Mex(Rep(G))) is isomorphic to the ﬁrst cohomology 1 ∗ group H (G, k ). Indeed, according to remark 4.1.1.3, the group π2(Mex(Rep(G))) coincides with ker Aut⊗ IdZ(Rep(G)) → Aut⊗ IdRep(G) .

Finally, we make a few remarks about the associator of the categorical group Mex(Rep(G)). This associator is a class in H3 H3(G, k∗), H2(G, k∗) . The last result of this section shows that it is the class of a certain crossed module.

Lemma 4.1.1.7 ([13], Lemma 2.7). The categorical group G associated with the crossed module ∂ Z3(G, k∗) ←− C2(G, k∗)/B2(G, k∗) (4.1)

is equivalent to Mex(Rep(G)).

Proof. Objects of the categorical group G are elements of Z3(G, k∗). A morphism between α, β ∈ Z3(G, k∗) is a 2-cochain c ∈ C2(G, k∗)/B2(G, k∗) such that ∂c = α · β−1. Define a functor F : G → Mex(Rep(G)) by F(α) = Z(G, α). For a morphism c : α → β define a braided tensor functor Id(c): V(G, α) → V(G, β), which is the identity functor with the tensor structure given by the 2-cochain c. Define F(c): Z(G, α) → Z(G, β) to be the functor Z(Id(c)) : Z(V(G, α)) → Z(V(G, β)) induced by Id(c).

By proposition 4.1.1.4 and lemma 4.1.1.5, the eﬀects of F on both π0 and π1 are

bijective. Hence F is an equivalence of categorical groups.

Remark 4.1.1.8. Similarly, Mex(Rep(G)) corresponds to the crossed-complex

∂ ∂ Z3(G, k∗) ←− C2(G, k∗) ←− C1(G, k∗) . 100

4.1.2 Invertible Objects of Z(G, α)

Let

n ∗ o P = (z, c) ∈ Z(G) × C1(G, k ) c(x)c(y) = α(x, y|z)c(xy), x, y ∈ G .

It is straightforward to see that the operation

(z, c)(w, d) = zw, cdα(−|z, w)−1 . (4.2)

makes P a group with the identity element53 (e, 1) and the inverses

− − − − (z, c) 1 = z 1, c 1α − z, z 1 .

Denote by

n 1 ∗ o Zα(G) = z ∈ Z(G) ∃c ∈ C (G, k ) such that c(x)c(y) = α (x, y|z) c(xy) ∀x, y ∈ G a subgroup of central elements of G.

Remark 4.1.2.1. Note that P and Zα(G) ﬁt in the short exact sequence

ι π Gb −→ P −→ Zα(G) , (4.3) where ι is the map χ 7→ (e, χ) and π is the canonical projection (z, c) 7→ z.

Recall that the Picard group Pic(C) of a monoidal category C is the group of isomorphism classes of invertible objects of C.

Theorem 4.1.2.2 ([13], Theorem 2.8). The Picard group Pic(Z(G, α)) is isomorphic to P.

Proof. The natural forgetful functor Z(G, α) → V(G, α) (forgetting about the braiding) induces a group homomorphism Pic(Z(G, α)) → Pic(V(G, α)) = G. However, because Z(G, α) is braided, this homomorphism factors through the center Z(G). Hence the

53 Here, 1 denotes the constant function 1(x) ≡ 1 ∈ k. 101

support of an invertible object I ∈ Pic(Z(G, α)) is a single central element z ∈ Z(G). The α-projective G-action54 on an invertible object I = I(z) is given by multiplication by c, which shows the correspondence on the level of sets. The group operation on Pic(Z(G, α)) is the tensor product in Z(G, α). The α-projective action on the tensor product I(z)⊗I(w) is diagonal and contributes a factor of α(−|z, w), which shows that the correspondence of sets is an isomorphism of groups.

The function q : P → Q/Z given by

q(z, c) = c(z) (4.4) is induced by the braiding on Z(G, α). It is a quadratic form on P. A 3-cocycle α ∈ Z3(G, k∗) is said to be soft if for any g ∈ G there is a c ∈ C1(G, k∗) such that c(x)c(y) = α(x, y|g)c(xy) for all x, y ∈ G.

Corollary 4.1.2.3 ([13], Corollary 2.9). The category Z(G, α) is pointed if and only if G is Abelian and α is soft.

Proof. The category Z(G, α) is pointed if and only if its group of invertible objects is of order |G|2. By theorem 4.1.2.2, we have

Gb = |G| = |Zα(G)| .

The ﬁrst equality is equivalent to G being Abelian, and the second is equivalent to α being soft.

Remark 4.1.2.4. For a pointed category Z(G, α), the group of objects P is a Lagrangian extension55 of Gb with respect to ι : Gb → P and the quadratic function (4.4).

54 See § 1.4.3. 55 See § 4.2. 102

To pursue our interest in pointed Drinfeld centers Z(G, α), we now assume that

3 ∗ G = B is Abelian. Denote by Zsoft(B, k ) the group of soft 3-cocycles. It is straightforward 3 ∗ 3 ∗ 3 ∗ to see that all 3-coboundaries are soft. Write Hsoft(B, k ) = Zsoft(B, k )/B (B, k ).

Lemma 4.1.2.5 ([13], Lemma 2.10). Let B be an Abelian group. Then the third soft

3 ∗ cohomology group Hsoft(B, k ) is the kernel of the alternation map 3 ∗ 3 ∗ Alt3 : H (B, k ) → HomZ Λ B, k .

Proof. The assignment α 7→ α(−, −|−) gives a homomorphism H3(B, k∗) → H2B, Map(B, k∗) into the second cohomology of B with coeﬃcients in the trivial B-module Map(B, k∗) of set-theoretic functions B → k∗. The soft cohomology

3 ∗ Hsoft(B, k ) is the kernel of this homomorphism. Since k∗ and Map(B, k∗) are divisible, the universal coeﬃcient formula implies that

the alternation map Alt2 gives an isomorphism H2(B, Map(B, k∗)) → Hom Λ2B, Map(B, k∗) . The following commutative diagram

3 ∗ 3 ∗ 2 ∗ 0 / Hsoft(B, k ) / H (B, k ) / H B, Map(B, k )

Alt3 Alt2 Hom Λ3B, k∗ / / Hom Λ2B, Map(B, k∗)

proves the result.

For an Abelian group B, deﬁne the category Mexpt(Rep(B)) to be the full categorical subgroup of Mex(Rep(B)) consisting of those modular extensions D of Rep(B) that are pointed categories. The computations above prove the following:

Theorem 4.1.2.6. The group of connected components π0 Mexpt(Rep(B)) = Mexpt(Rep(B)) ﬁts into the short exact sequence

3 3 {0} −→ Mexpt(Rep(B)) −→ H (B, Q/Z) −→ HomZ Λ B, Q/Z −→ {0} 103

4.2 Lagrangian Extensions of Abelian Groups

4.2.1 The Categorical Group Lex(B)

Let B and D be Abelian groups. An extension of B by D is a short exact sequence of Abelian groups D −→ A −→ B .

Denote by the Baer sum of extensions:

0 0 ι π ι 0 π i 0 ¯ p D −→ A −→ B D −→ A −→ B = D −→ A ×B A /∆(D) −→ B , (4.5)

0 where A ×B A is the ﬁbered product, ∆¯ (D) is the antidiagonal subgroup, i(d) = (ι(d), 0) = (0, ι0(d)), and p(a, a0) = π(a) = π0(a0). The Baer sum is both associative and commutative, and the Baer sum of extensions is again an extension56. Let B be an Abelian group. A Lagrangian extension of B is a triple (A, q, ι), where A

is an Abelian group, ι : B ,→ A is an embedding of groups, and q : A → Q/Z is a nondegenerate quadratic function such that ι[B] A is a Lagrangian subgroup with respect to q. The collection of Lagrangian extensions of a given Abelian group B forms a category which we denote by Lex(B). A morphism (A, q, ι) → (A0, q0, ι0) in Lex(L) is a homomorphism ω : A → A0 such that q0 ◦ ω = q and ω ◦ ι = ι0.

0 0 0 0 0 0 Let (A, q, ι), (A , q , ι ) ∈ Lex(B). Deﬁne the sum (A, q, ι) (A , q , ι ) by

0 0 0 0 ¯ 0 (A, q, ι) (A , q , ι ) = A ×bB A /∆(B), q q , δ ,

where × 0 0 ∈ × 0 0 0 − 0 0 0 ∀ ∈ A bB A = (a, a ) A A q(a + ι(x)) + q (a ι (x)) = q(a) + q (a ) x B is the ﬁbered product, and ∆¯ (B) = {(ι(b), −ι0(b))|b ∈ B}

56 See § 4.3 for more information. 104

0 is the antidiagonal subgroup. Deﬁne a quadratic functionq ˜ : A × A → Q/Z by 0 0 0 q˜(a, a ) = q(a) + q (a ). The kernel ofq ˜| × 0 is precisely ∆¯ (B), and hence the restriction A bBA 0 ¯ 0 descends to the quotient A ×bB A /∆(B) and gives a nondegenerate function q q on the quotient. Indeed, the orthogonal complement ∆¯ (B)⊥ in A × A0 (with respect toq ˜) is the

0 0 0 ﬁbered product A ×bB A : for (a, a ) ∈ A × A , the orthogonality condition σ˜ (a, a), (ι(x), −ι0(x)) = 0 with (ι(x), −ι0(x)) is equivalent to σ(a, ι(x)) = σ0(a0, ι0(x)), or 0 0 57 0 ¯ simply π(a) = π (a ). Finally, the embedding δ : B → A ×bB A /∆(B) is given by δ(b) = (ι(b), 0) = (0, ι0(b)).

Symmetry of the operation is clear. Associativity of follows from associativity of the Baer sum (4.5). This works because the Baer sum is compatibile with the quadratic structure on the extensions A and A0.

By the trivial Lagrangian extension, we mean (B × bB, qstd, ι) with qstd(b, χ) = χ(b) and ι(b) = (b, 0). The trivial Lagrangian extension is the monoidal unit object in Lex(B). That is, Lex(B) is a symmetric monoidal groupoid. Deﬁne the conjugate of the Lagrangian extension (A, q, ι) by (A, q, ι) = (A, −q, ι). The conjugate is the dual object in Lex(B). This makes Lex(B) a symmetric categorical group with respect to . Now we shall discuss the functoriality of Lex(B). Namely, for a homomorphism of Abelian groups f : B → D, we will deﬁne a functor between symmetric categorical groups Lex(B) → Lex(D). For a Lagrangian extension (A, q, ι) of B, set

(A × Db) × D A0 = bB . ∆(B)

57 Here, σ and σ0 are the symmetric bilinear forms associated to q and q0, respectively. 105

Note that A0 ﬁts into the commutative diagram with exact rows and columns

ι0 . π0 D / A × Db × D ∆(B) / Db O bB O f

ι×0 pr2 B / A ×bB Db / Db

bf B ι / A π / bB Now deﬁne a Lagrangian extension of D   (A ×bB Db) × D 0 0 Lex( f )(A, q, ι) =  , q , ι  , (4.6)  ∆(B) 

0 0 0 where q : A → Q/Z is deﬁned as follows. The formula q (a, χ, d) = q(a) + χ(d) deﬁne a degenerate quadratic function on (A ×bB Db) × D with kernel ∆(B).

Remark 4.2.1.1. Suppose now that f is surjective, and let K = ker( f ). It can be veriﬁed

⊥ that in this case A ×bB Db coincides with the orthogonal complement K of K in A. (A × Db) × D Moreover, the quotient bB coincides with the isotropic contraction K⊥/K along ∆(B) K. This contains B/K, which is isomorphic to D. Thus (4.6) reduces to

⊥ 0 Lex( f )(A, q, ι) = K /K, q|K⊥ , ι ,

where ι0 : D ' B/K → K⊥/K is induced by ι : B → K⊥.

Deﬁne a functor Lex(B) → Mexpt(C(B)) by

(A, q, ι) 7→ C(A, q) . (4.7)

Proposition 4.2.1.2 ([13], Proposition 3.3). The functor (4.7) is an equivalence of groupoids.

Proof. Indeed, any pointed modular extension of C(B) has the form C(A, q) for some Lagrangian extension (A, q) of B. Any equivalence of extensions C(A, q) → CA0, q0

0 0 correponds to an isomorphism of Lagrangian extensions (A, q) → (A , ι ). 106

The equivalence (4.7) allows us to transfer the monoidal structure C(B) from

Mexpt(C(B)) to Lex(B). Note ﬁrst that the etale algebra R = F(I) ∈ C(B) C(B) corresponds, under the standard identiﬁcation C(B) C(B) 'C(B⊕B), to the antidiagonal subgroup δ¯(B) = {(b, −b)| b ∈ B} ≤ B⊕B. Indeed, the tensor product functor

C(B) C(B) → C(B) corresponds to the direct image of the addition homomorphism a : B × B → B, (x, y) 7→ x + y. Thus R is the inverse image a∗(I) = R(δ¯(B)). Now, the chain of braided equivalences

0 0 0 0 loc 0 0 loc C(A, q) C C(A , q ) = (C(A, q) C(A , q )) 'C(A × A , q × q ) ' (B) R R(δ(B)) 'C δ(B)⊥/δ(B), q × q0

0 0 0 0 0 identiﬁes the product C(A, q) C(B) C(A , q ) with the sum (A, q, ι) (A , q , ι ). The equivalence (4.7) implies that the group Lex(B) of Lagrangian extensions is isomorphic to Mexpt(C(B)). Since C(B) 'Rep(bB) as symmetric fusion categories, we have:

Corollary 4.2.1.3. The group Lex(B) of Lagrangian extensions of B is isomorphic to the group Mexpt Rep(bB) .

Proof. Follows immediately from theorem 4.1.2.6 and proposition 4.2.1.2.

Corollary 4.2.1.4. The group Lex(bB) ﬁts in the short exact sequence

3 3 {0} −→ Lex(bB) −→ H (B, Q/Z) −→ HomZ Λ B, Q/Z −→ {0}

Proof. Follows immediately from theorem 4.1.2.6, proposition 4.2.1.2, and corollary

4.2.1.3.

Remark 4.2.1.5. Here is an alternative description of the group Lex(bB). In [3], it was proved that for a ﬁnite Abelian group B, there is a short exact sequence

3 Z S 2 {0} −→ Λ B −→ H3(B, Z) −→ Tor1 (B, B) −→ {0} 107

Z S 2 Z Here, Tor1 (B, B) is the subgroup of Tor1 (B, B) consisting of S 2-anti-invariants. 58 Applying the contravariant functor HomZ(−, Q/Z) to this sequence yields Z S 2 3 {0} → HomZ Tor1 (B, B) , Q/Z → HomZ(H3(B, Z), Q/Z) → HomZ Λ B, Q/Z → {0}

By the Universal Coeﬃcient Theorem59, there is a short exact sequence

{ } −→ 1 (H B, , / ) −→ H3 B, / −→ (H B, , / ) −→ { } 0 ExtZ 2( Z) Q Z ( Q Z) HomZ 3( Z) Q Z 0

/ 60 1 (M, / ) Since Q Z is a divisible group, it is an injective Z-module, whence ExtZ Q Z is trivial for any Abelian group M. Thus the above short exact sequence gives an

3 isomorphism H (B, Q/Z) ' HomZ(H3(B, Z), Q/Z). The commutative diagram with short exact rows

3 3 Lex(bB) / H (B, Q/Z) / HomZ Λ B, Q/Z

∼ Z S 2 3 HomZ Tor1 (B, B) , Q/Z / HomZ(H3(B, Z), Q/Z) / HomZ Λ B, Q/Z

Z S 2 shows that Lex(bB) is isomorphic to the dual of the group Tor1 (B, B) .

Remark 4.2.1.6. For any positive integer n, the third cohomology group

3 3 H (Z/nZ, Q/Z) ' Z/nZ. Since Z/nZ is cyclic, Λ (Z/nZ) = {0}. Therefore the short exact sequence

3 3 {0} −→ Lex(Z/nZ) −→ H (Z/nZ, Q/Z) −→ HomZ Λ (Z/nZ), Q/Z −→ {0}

gives an isomorphism Lex(Z/nZ) ' Z/nZ.

Example 4.2.1.7. Here, we explicitly compute the group of Lagrangian extensions

Lex(Z/2Z). Since a Lagrangian extension A )Z/2Z is an Abelian group extension of 2 Z/2Z by Z/2Z, A is of order 4, and either A ' (Z/2Z) or A ' Z/4Z. 58 Here, we implicitly use the fact that Q/Z is an injective Z-module, and so its contravariant Hom functor preserves short exact sequences. 59 See, e.g.,[22]. 60 For the basic facts on injective modules, see, e.g.,[22, § IV.3]. 108

2 Let A = (Z/2Z) . The matrices of possible nondegenerate symmetric bilinear forms σ : A × A → Q/Z such that the ﬁrst summand Z/2Z is Lagrangian are      1   1   0   0   2   2    and   .  1 1   1     0  2 2 2

Here, entries are in Q/Z. For the ﬁrst one, a compatible quadratic function q± : A → Q/Z has the form 1 1 q (e ) = 0, q (e ) = ± , q (e + e ) = ∓ . ± 1 ± 2 4 ± 1 2 4 The corresponding classes in Lex(Z/2Z) are isomorphic and nontrivial. The second bilinear form corresponds to the trivial Lagrangian extension Z/2Z × Z[/2Z, qstd, canonical inclusion .

It follows from the following example 4.2.1.8 that the group A = Z/4Z is not realizable as a Lagrangian extension of Z/2Z.

` Example 4.2.1.8. More generally, let B = Z/2 Z for some positive integer `. As in ` example 4.2.1.7 above, a Lagrangian extension A )Z/2 Z is an Abelian group extension ` ` 2` 2` of Z/2 Z by Z/2 Z, whence |A| = 2 . Suppose that A = Z/2 Z, and consider the short exact sequence

` · 2 (·) [ ]2` {0} / Z/2`Z / Z/22`Z / Z/2`Z / {0}

Because subgroups of ﬁnite cyclic groups of a given order are unique, a Lagrangian

2` subgroup of the middle term is unique if it exists. A quadratic function q : Z/2 Z → Q/Z is determined by its value at 1, and this must be of the form

k q(1) = 22`+1

` for some integer k. Since the image of Z/2 Z must be Lagrangian with respect to such q, ` 2` we have q 2 ≡ 0 (mod Z), which implies that 2 q(1) ≡ 0 (mod Z). Thus 109

22`k k ≡ 0 (mod ), and so ≡ 0 (mod ). Therefore k is even. Write k = 2m for some 22`+1 Z 2 Z integer m. We have m q(1) = . 22` Now m σ(1, 1) = q(2) − 2q(1) = 4q(1) − 2q(1) = 2q(1) = . 22`−1 But then m σ 22`−1, 1 = 22`−1σ(1, 1) = 22`−1 = m ≡ 0 (mod ), 22`−1 Z

2`−1 2` so 0 , 2 ∈ ker(σ) and hence σ is degenerate. Therefore Z/2 Z is not realizable as a ` Lagrangian extension of Z/2 Z.

2`−1 Example 4.2.1.9. Let now A = Z/2 Z × Z/2Z. Set e1 = (1, 0), e2 = (0, 1), so that D `−1 E A = he1, e2i, and set L = 2 e1 + e2 . We wish to determine possible quadratic functions

q : A → Q/Z with respect to which the subgroup L A is Lagrangian. Since any element

~x ∈ A has the form ~x = ae1 + be2 for some a, b ∈ Z, such a function q has the form

2 2 q(a, b) = a q(e1) + b q(e2) + abσ(e1, e2), (4.8)

where σ : A × A → Q/Z is the symmetric bilinear form associated to q. In order that q be well-deﬁned, we must have q a + 22`−1, b = q(a, b) and q(a, b + 2) = q(a, b) for all (a, b) ∈ A. Therefore

2` 2 q(e1) = 0, 4q(e2) = 0, 2σ(e1, e2) = 0. (4.9)

The quadratic condition on such q requires that

q((a, b) + (c, d)) = q(a, b) + q(c, d) + σ((a, b) , (c, d)) ∀ (a, b) , (c, d) ∈ A.

Therefore

2q(ei) = σ(ei, ei), i ∈ {1, 2}. (4.10) 110

Combining the conditions (4.9) and (4.10), we may set

1 1 q(e ) = , q (e ) = ± . 1 22` 2 4

To remove the indeterminacy in q(e2), we apply the Lagrangian condition. Since the

` 2 subgroup L A has order 2 , |L| = |A|. Therefore L is Lagrangian with respect to q if and only if L is isotropic with respect to q. Isotropy of L amounts to the condition

2`−1 q 2 e1 + e2 = 0.

By (4.8),

`−1 2`−2 q 2 e1 + e2 = 2 q(e1) + q(e2) 22`−2 1 1 1 1 =⇒ ± = ± = 0 ⇐⇒ q(e ) = − . 22` 4 4 4 2 4 Therefore (4.8) becomes

a2 b2 q(a, b) = − + abσ(e , e ). (4.11) 22` 4 1 2 ( ) 1 It remains to determine σ(e , e ). By (4.9), σ(e , e ) ∈ 0, . We consider two cases. 1 2 1 2 2

Case I: σ(e1, e2) = 0. Then σ has the matrix representation  1   0   2`−1      .    1   0  2 Let (a, b) ∈ ker(σ). Then σ((a, b), (x, y)) = 0 for all (x, y) ∈ A. By bilinearity of σ, this condition amounts to

axσ(e1, e1) + byσ(e2, e2) = 0 ∀(x, y) ∈ A,

viz., ax by + ≡ 0 (mod ) ∀(x, y) ∈ A. 22`−1 2 Z 111 b In particular, for (x, y) = (0, 1), we have ≡ 0 (mod ), i.e., b ≡ 0 (mod 2). Similarly, 2 Z setting (x, y) = (1, 0), we get that a ≡ 0 mod 22`−1 , and therefore (a, b) = (0, 0). Therefore σ is nondegenerate.

In this case, we can deﬁne q : A → Q/Z according to the formula (4.11): a2 b2 q(a, b) = − . 22` 4 1 Case II: σ(e , e ) = . Then σ has the matrix representation 1 2 2  1 1     2`−1 2      .    1 1    2 2 Let (a, b) ∈ ker(σ). Then by bilinearity of σ, we have ax ay bx by + + + ≡ 0 (mod ) ∀(x, y) ∈ A. 22`−1 2 2 2 Z In particular, for (x, y) = (0, 1), we have a + b ≡ 0 (mod ) , 2 Z so that either both a and b are even or both a and b are odd. Setting (x, y) = (1, 0) yields a b + ≡ 0 (mod ) . (4.12) 22`−1 2 Z Assume by way of contradiction that a and b are both odd. Write a = 1 + 2a0, b = 1 + 2b0

0 0 for some a , b ∈ Z. Substituting these values into (4.12) yields 1 + 2a0 1 + 2b0 + ≡ 0 (mod ) . (4.13) 22`−1 2 Z Rewriting (4.13) in terms of the common denominator 22`−1 yields 1 + 2a0 + 22`−2 + 22`−1b0 ≡ 0 (mod ) . (4.14) 22`−1 Z Since the LHS of (4.14) is an integer, its numerator must be even, yet its numerator is clearly odd61 (⇒⇐). Therefore we cannot have that a and b are odd, so we must have that

61 Here, we assume without loss of generality that ` , 1, since example 4.2.1.7( q.v.) already disposed of that case. 112

a and b are even. But b even means that b ≡ 0 (mod 2), i.e., b = 0, so by (4.12),

a ≡ 0 (mod ) , 22`−1 Z i.e., a ≡ 0 mod 22`−1 . Therefore a = 0 as well, and therefore σ is nondegenerate.

In this case, we can deﬁne q : A → Q/Z according the the formula (4.11):

a2 b2 ab q(a, b) = − + . 22` 4 2

`−1 In both cases, the subgroup L A generated by the pair 2 , 1 is Lagrangian with ` respect to q and is isomorphic to Z/2 Z.

4.3 The Categorical Group Ext(B, D)

Let B and D be Abelian groups. Denote by Ext(B, D) the category whose objects are ι π Abelian group extensions D −→ A −→ B and whose morphisms are group homomorphisms ξ : A → A0 making the diagram

A ? ι π {0} / D ξ B / {0} ?

ι0 π0 A0

commute. Note that such ξ is necessarily an isomorphism, i.e., Ext(B, D) is a groupoid. The category Ext(B, D) is monoidal with respect to the Baer sum (4.5). The unit object in Ext(B, D) is the trivial extension

D −→ D × B −→ B , 113

with the canonical embedding and projection maps. Associativity of the operation is given by the following diagram:

i1 0 00 . . p1 D / A ×B A ×B A ∆¯ (D) ∆¯ (D) / B

0 . 00. D / A ×B A ∆¯ (D) ×B A ∆¯ (D) / B i2 p2 Consider the triple ﬁbered product

0 00 n 0 00 0 00 0 0 00 00 o A ×B A ×B A = a, a , a ∈ A × A × A π(a) = π (a ) = π (a ) .

0 00 Deﬁne a subgroup ∇(D) ≤ A ×B A ×B A by

n 0 00 0 o ∇(D) = 0, ι (x), −ι (x) + ι(y), ι (y), 0 x, y ∈ D .

0 00 0 00 . . There are natural maps A ×B A ×B A ∇(D) → A ×B A ×B A ∆¯ (D) ∆¯ (D) and

0 00 0 . 00. A ×B A ×B A ∇(D) → A ×B A ∆¯ (D) ×B A ∆¯ (D) that commute with α. We equip the product A × A0 × A00 with the quadratic functionq ˜ = q ⊕ q0 ⊕ q00 and its associated bilinear formσ ˜ = σ ⊕ σ0 ⊕ σ00, and consider the orthogonal complement ∇(D)⊥ in

0 00 ⊥ 0 00 0 00 ⊥ A × A × A . We claim that ∇(D) = A ×B A ×B A . Let a, a , a ∈ ∇(D) . Then for any x, y ∈ D, we have

σ˜ a, a0, a00 , ι(x), ι0(x), 0 = 0 = σ˜ a, a0, a00 , 0, ι0(y), ι00(y) .

The ﬁrst equality implies that σ(a, ι(x)) = σ0 a0, ι0(x) for any x ∈ D, i.e., π(a) = π0(a0). The second equality implies that σ0 a0, ι0(y) = σ00 a00, ι00(y) for any y ∈ D, i.e.,

0 0 00 00 0 00 0 00 π (a ) = π (a ). It follows that a, a , a ∈ A ×B A ×B A . It follows that the operation ι π is associative up to isomorphism. The opposite extension to D −→ A −→ B is the ι −π extension D −→ A −→ B. Note that the Baer sum

ι π ι −π D −→ A −→ B D −→ A −→ B 114

is isomorphic to the trivial extension. Altogether, this makes the category Ext(B, D) into a categorical group with respect to . The standard invariants of the categorical group Ext B, D π Ext B, D 1 B, D π Ext B, D B, D ( ) are 0( ( )) = ExtZ( ) and 1( ( )) = HomZ( ). Ext is functorial in each of its arguments; it is contravariant in its ﬁrst argument and covariant in its second argument.

4.3.1 Wherein the Arguments of Ext are Dual to One Another

In the special case when the arguments B and D are dual to each other, the categorical group Ext(B, D) has a natural symmetry. Namely, deﬁne the contravariant functor

T : Ext(bB, B) → Ext(bB, B) by ! ι π bπ◦evB ι T B / A / bB = B / Ab b / bB ,

b where evB : B → bB is the canonical evaluation isomorphism. The functor T is an involutive autoequivalence. More precisely, we have a natural isomorphism i : Id → T 2 b −1 (given by evA : A → Ab) such that iT(X) = T(iX) . Namely, let X ∈ Ext(bB, B) be the ι π Lagrangian extension B / A / bB , so that T(X) is the extension

π ι bB b / Ab b / bB .

Deﬁne a map X → T 2(X) by the following diagram:

ι π B / A / bB . O

evB evA evdB b bB / bAb / bB bι bπ The left-hand square of this diagram commutes due to naturality of the evaluation map b −1 ev(·) : Id → (b·). Note that evbB = evcB . Indeed, evbB : χ 7→ (ev(a) 7→ ev(a)(χ) = χ(a)) for

a ∈ B, whence evcB ◦ evbB : χ 7→ (a 7→ χ(a)) for a ∈ B. This implies that the right-hand square of the diagram also commutes, and therefore the extensions X and T 2(X) are equivalent. 115

More generally, let G be a categorical group. For any X ∈ G, there is a map

AutG(I) → AutG(X) deﬁned as follows. For α ∈ AutG(I), deﬁne a ∈ AutG(X) by the diagram

λX I⊗X / X

α⊗IdX a I⊗X / X λX Here, λ denotes the left unit isomorphism in G. If G is a categorical group, then the above map is an isomorphism. Let gX : AutG(X) → AutG(I) denote the inverse of this map. Let T : G → G be a contravariant monoidal autoequivalence with i : Id → T 2 a

−1 natural isomorphism such that iT(X) = T(iX) for any X ∈ G. We say that an object X ∈ G is T-equivariant if there is an isomorphism x : X → T(X) making the following diagram commute:

iX X / T 2(X) (4.15)

x T(x) ! { T(X) τ Let τ be the eﬀect of T on π0(G), and denote by π0(G)0 the subgroup of classes of T-equivariant objects.

Proposition 4.3.1.1 ([13], Proposition 3.8). The sequence

τ τ 1 {0} / π0(G)0 / π0(G) / H (Z/2Z, π1(G))

is exact.

Proof. We say that the class of an object X ∈ G is τ-invariant if there is a morphism

x : X → T(X). Deﬁne a = a(x) ∈ AutG(X) as the counterclockwise composition of the diagram (4.15). Let α(x) ∈ AutG(I) = π1(G) be the corresponding automorphism of the monoidal unit I. Then we have τ(α) = T(α) = T(a). Moreover, for any Y ∈ G and any

−1 morphism x : X → Y, we have gX(a) = gY xax . 116

Applying the functor T to the diagram (4.15) gives

T(iX) T(X) o T 3(X) (4.16) c :

T(x) T 2(x) T 2(X)

Note that T(a) ∈ AutG(T(X)) is the clockwise composition of the diagram (4.16). We claim that T(a) = xa−1 x−1 (4.17)

−1 In terms of elements of AutG(I) we have τ(α) = α . Consider the diagram

x X / T(X) (4.18) O 4 T(iX)

i T 3(X) u T(X) iX : a T(a) 2 T (x) T 2(X)

T(x) ) X x / T(X)

The rightmost cell of (4.18) commutes by definition of T(a). The bottom left cell commutes by definition of a. The top center cell commutes by naturality of i, and so the diagram commutes overall. (4.17) follows from commutativity of (4.18). Now let X ∈ G be τ-invariant with an isomorphism x : X → T(X). Let x0 : X → T(X) be another isomorphism. Define b ∈ Aut(X) by the diagram

x X / T(X) _ =

b x0 X 117

It follows from the diagram

a(x0) X / X <

b a(x) " xT(b)x−1 X o X

0 x x iX } T(X) x h

T(b) T(x) Ö v T(X) o T 2(X) T(x0)

0 −1 −1 0 −1 that a(x ) = a(x)xT(b) x b. This translates to α(x ) = α(x)τ(β) β, where β ∈ AutG(I) is

the automorphism corresponding to b ∈ AutG(X). Thus we have a map

τ 1 π0(G) → H (Z/2Z, π1(G)) ,

τ sending X ∈ π0(G) to the class of α(x) in

1 −1 −1 H (Z/2Z, π1(G)) = {α ∈ π1(G) |τ(α) = α }/{βτ(β) | β ∈ π1(G)} .

τ Now we check that this map is a group homomorphism. Given X, Y ∈ π0(G) , choose x : X → T(X) and y : Y → T(Y). Then

x⊗y TX,Y X⊗Y / T(X)⊗T(Y) / T(X⊗Y) 118

is a weak equivariance structure for X ⊗ Y. Compute a(x⊗y) = a(x|y) according to the following diagram:

iX⊗Y

+ X⊗Y T 2 (X ⊗ Y)

T(TX,Y )

iX⊗iY

x T (T(X)⊗T(Y)) x⊗y TT(X),T(Y) v T 2(X)⊗T 2(Y) x|y T(x)⊗T(y) % T(x⊗y) T(X)⊗T(Y) T(x|y) O TX,Y ( z T (X⊗Y) q

It follows from the above diagram that a(x|y) = a(x)⊗a(y). Finally, by the construction, the class of α(x) is trivial if and only if a τ-invariant structure on X can be promoted to a

T-equivariant one.

T π Ext B, B 1 B, B The eﬀect of on 0( (b )) = ExtZ(b ) is an involution τ π T 1 B, B → 1 B, B B B = 0( ) : ExtZ(b ) ExtZ(b ), which sends an extension of b by to its dual. Upon identiﬁcation of the double dual with B, the dual extension becomes another

extension of bB by B.

62 If we identify HomZ(bB, B) with the tensor square B⊗B, then the eﬀect of T on

π1(Ext(bB, B)) becomes the permutation involution.

Remark 4.3.1.2. The above proposition gives an exact sequence

{0} / Ext1 (B, B)τ / Ext1 (B, B)τ / H1( /2 , B⊗B) Z b 0 Z b Z Z

62 Using the fact that B is ﬁnite. 119

The last group is

1 H (Z/2Z, B⊗B) = {α ∈ B⊗B | τ(α) = −α}/ {β − τ(β) |β ∈ B⊗B} ,

63 1 with τ acting as the transposition of tensor factors. Deﬁne a map B2 → H (Z/2Z, B⊗B) by b 7→ b ⊗ b. This map is an isomorphism. To see that, ﬁrst note that the functor

1 1 B 7→ H (Z/2Z, B⊗B) is linear. By linearity, it suﬃces to show that B2 → H (Z/2Z, B⊗B) is an isomorphism for cyclic B. This is obviously true when B is of odd order, since both

1 ` B2 and H (Z/2Z, B⊗B) are equal to {0} in that case. For B = Z/2 Z, we have

Z1 /2 , /2` ⊗ /2` = /2` and B1 /2 , /2` ⊗ /2` = {0} , Z Z Z Z Z Z Z Z 2 Z Z Z Z Z Z

1 whence the map B2 → H (Z/2Z, B⊗B) is indeed an isomorphism. Altogether, this gives an exact sequence

{0} / Ext1 (B, B)τ / Ext1 (B, B)τ / B Z b 0 Z b 2

4.4 Comparing Lex(B) with Ext(bB, B)

4.4.1 F ’n’ φ

Deﬁne a functor F : Lex (B) → Ext(bB, B) by

ι π F(A, q, ι) = B −→ A −→ bB ,

where (π(a)) (x) = q(a + ι(x)) − q(a) ∀a ∈ A, x ∈ B. Note that F is monoidal with the obvious monoidal structure. F φ B → 1 B, B The functor induces a homomorphism of Abelian groups : Lex( ) ExtZ(b ). First, we examine the image of φ.

Proposition 4.4.1.1 ([13], Proposition 3.10). The extension F(A, q, ι) is T-equivariant. 63 Note that the condition 2b = 0 implies that τ(b⊗b) = −b⊗b. 120

Proof. Deﬁne f : A → Abby f (a) = σ (a, −) for a ∈ A. Here, σ is the polarization of q. Such f makes the diagram

ι π B / A / bB

e f bB / Ab / bB bπ bι commute. Indeed, for x ∈ B and a ∈ A, we have

( f (ι(x))) (a) = σ (ι(x), a) = (π(a)) (x) = e(x) (π(a)) = bπ(e(x))(a), and (π(a)) (x) = σ(a, x) = ( f (a))(x) = bι ( f (a)) (x). It is straightforward to check that f is a

T-equivariant structure on F(A, q, ι).

In other words, proposition 4.4.1.1 shows that the homomorphism φ lands in 1 B, B τ φ 1 B, B τ K B C B ExtZ(b ) , with the image of being precisely ExtZ(b )0. Denote by ( ) and ( ) the kernel and the cokernel, respectively, of the homomorphism φ B → 1 B, B τ K C : Lex( ) ExtZ(b ) . Clearly, and are functors from the category of ﬁnite Abelian groups to itself that ﬁt into the exact sequence

φ {0} / K(B) / Lex(B) / Ext1 (B, B)τ / C(B) / {0} Z b

Next, we compute the polarization of the functor Lex. Note that for an extension

ι π {0} / B / E / Db / {0}

the embedding64 B ⊕ D ,→ E ⊕ Eb is a Lagrangian subgroup of the quadratic group

(E ⊕ Eb, qstd). In other words, (E ⊕ Eb, qstd) becomes a Lagrangian extension of B⊕D. This υ 1 D, B → B ⊕ D allows us to deﬁne a map : ExtZ(b ) Lex( ) by

ι π υ B −→ E −→ Db = E ⊕ Eb, qstd, ι .

64 −1 The direct sum of ι and bπ ◦ ev . 121

Proposition 4.4.1.2 ([13], Proposition 3.11). The sequence

υ κ { } −→ 1 D, B −→ B ⊕ D −→ B ⊕ D −→ { } , 0 ExtZ(b ) Lex( ) Lex( ) Lex( ) 0 where κ : Lex(B ⊕ D) → Lex(B) ⊕ Lex(D) is the canonical mapping, is exact.

Proof. By remark 4.2.1.1, κ maps the class of a Lagrangian extension (A, q, ι) of B ⊕ D into the pair of Lagrangian extensions

⊥ ⊥ B ( D /D, D ( B /B .

For (A, q, ι) in the kernel of κ, we have

⊥ ⊥ B ( D /D ' B ⊕ bB, D ( B /B ' D ⊕ Db , where B (respectively D) is embedded as the ﬁrst summand and is Lagrangian with respect to the standard quadratic function on B ⊕ bB (respectively D ⊕ Db).

Now we want show that the Lagrangian extension (E ⊕ Eb, qstd) of B⊕D (the eﬀect of the map υ on an extension B → E → Db) is always in the kernel of κ. Indeed, the orthogonal complement of D in E ⊕ Eb has the form ⊥ D = (e, ε) ∈ E ⊕ Eb σstd (e, ε), (0,bπ(d) = 0 ∀d ∈ D = B ⊕ Eb .

Furthermore,

⊥ . D D ' B ⊕ Eb 0 ⊕ bπ[D] ' B ⊕ bB .

Analogous results hold for B⊥ B. This shows that im(υ) ⊆ ker(κ). To see that im(υ) = ker(κ), we show that, for (A, q, ι) ∈ ker(κ), there exists an ι π extension B −→ E −→ Db of Db such that

(A, q, ι) ' E ⊕ Eb, qstd, ι ⊕ bπ .

We start by extracting such E out of A. Since (A, q, ι) is in the kernel of κ, the isotropic contraction along B is trivial as a Lagrangian extension of D: B⊥/B ' D ⊕ Db. By lifting 122

⊥ ⊥ the corresponding projections p1 : B /B D, p2 : B /B Db, we obtain surjections ⊥ ⊥ ep1 : B D and ep2 : B Db. Now deﬁne E to be ker(ep1). Since ker(ep1) ∩ ker(ep2) = B, the sequence ι p B −→ E −→e2 Db (4.19)

⊥ ⊥ is short exact. Similarly, D /D ' B ⊕ bB, and so we have projections w1 : D /D B, ⊥ ⊥ ⊥ w2 : D /D bB and corresponding surjections we1 : D B, we2 : D bB. Deﬁne 0 E = ker(we1). We then have the analogous short exact sequence

ι0 w D −→ E0 −→e2 bB . (4.20)

By their deﬁnitions, E and E0 are subgroups of A. We show that A is their direct sum. Since the orders of E and E0 square to the order of A, all we need to show is that E ∩ E0 = 0. Let x ∈ E ∩ E0. Then x ∈ B⊥ ∩ D⊥ = (B ⊕ D)⊥ = B ⊕ D since B ⊕ D is

Lagrangian in A. Since ep1|B⊕D is the second projection and we1|B⊕D is the ﬁrst,

ep1(x) = we1(x) = 0 implies that x = 0. Now we show that E and E0 are Lagrangian subgroups of the quadratic group A. Together with the decomposition E⊕E0 = A, that will give us a nondegenerate pairing

between E and E0, identifying E0 with Eb. What we are actually going to show is that E is isotropic; this is suﬃcient, since |E|2 = |A|. Recall that the kernel of the restriction of q to

⊥ ⊥ B is B, so for any x ∈ B we have q(x) = qB⊥/B(x ¯), wherex ¯ is the coset of x modulo B.

Now for x ∈ E = ker(ep1), the cosetx ¯ lies in ker(p1), which is an isotropic subgroup of ⊥ 0 B /B. Hence q(x) = qB⊥/B(x ¯) = 0. An analogous argument shows that E is also Lagrangian. Therefore since E⊕E0 = A, we must have that E and E0 are dual to one

0 another. More precisely, the pairing E ⊗E → Q/Z given by (y, x) 7→ q(y + x) is ∼ nondegenerate and gives an isomorphism of quadratic groups E0 −→ Eb. 123

The last thing we need to check is that, upon identiﬁcation between E0 and Eb, the

0 extension (4.20) is the dual of (4.19). In other words, we needbι = we2 and bι = ep2, which

follows immediately from the deﬁnitions.

Corollary 4.4.1.3 ([13], Corollary 3.12). The functors K and C are additive.

Proof. B 7→ 1 B, B τ 1 D, B The polarization of the functor ExtZ(b ) is ExtZ(b ). Moreover, the natural transformation φ induces an isomorphism of polarizations. Finally, note that

K({0}) = C({0}) = {0}.

Next, we describe the kernel functor K.

Lemma 4.4.1.4 ([13], Lemma 3.13). The functor K ﬁts into the exact sequence

{0} / ∧2B / B⊗B / Q(bB, Q/Z) / K(B) / {0} ,

where ∧2B is the subgroup of antisymmetric elements of B⊗B.

Proof. Let (A, q, ι) ∈ K(B). Then, as an Abelian group, A can be identiﬁed with B × bB. Moreover, under this identiﬁcation, ι becomes the canonical inclusion. Furthermore, the quadratic function q is such that

q(b + b0, χ) − q(b0, χ) = π(b0, χ)(b) = χ(b) .

Denote byq ˜ the restriction of q to bB. Then for b ∈ B, χ ∈ bB, we have

q (b, χ) = χ(b) + q˜ (χ) .

Noting that χ(b) = qstd(b, χ), one can write

(A, q, ι) ' (B × bB, qstd + q˜, ι)

as Lagrangian extensions of B. In other words, the map Q(bB, Q/Z) → K(B) given by

q˜ 7→ (B × bB, qstd + q˜, ι) is an epimorphism of groups. 124

Note that the kernel of the map Q(bB, Q/Z) → K(B) is the image of the map B⊗B → Q(bB, Q/Z) sending r to the quadratic function q(χ) = (χ⊗χ)(r). Clearly such q is zero if and only if τ(r) = −r, where τ : B⊗B → B⊗B is the transposition automorphism.

Corollary 4.4.1.5 ([13], Corollary 3.14). The functor K is isomorphic to the functor B 7→ B/2B.

Proof. Deﬁne a map Q(bB, Q/Z) → B/2B by assigning to q an element x ∈ B/2B such that 2q(χ) = χ(x) for all χ ∈ B[/2B. It is straightforward that this map is zero on the image of

B⊗B → Q(bB, Q/Z), thus giving the map K(B) → B/2B. Finally, by looking at the eﬀect on a cyclic 2-group B, we can see that the map K(B) → B/2B is an isomorphism.

In the rest of the section, we examine the cokernel functor C.

` Remark 4.4.1.6. Note that C(Z/2 Z) ' Z/2Z. Indeed, a generator of 1 / ` , / ` ' / ` ExtZ(Z 2 Z Z 2 Z) Z 2 Z has the form

` 2` ` Z/2 Z −→ Z/2 Z −→ Z/2 Z

2` ` Example 4.2.1.8 shows that Z/2 Z is not realizable as a Lagrangian extension of Z/2 Z. / ` → 1 / ` , / ` Thus the map Lex(Z 2 Z) ExtZ(Z 2 Z Z 2 Z) is not surjective. Note also that any extension of the form

` 2`−1 ` Z/2 Z −→ Z/2 Z × Z/2Z −→ Z/2 Z (4.21)

1 / ` , / ` ' / ` is twice a generator of ExtZ(Z 2 Z Z 2 Z) Z 2 Z. Now, example 4.2.1.8 shows that ` the middle term of (4.21) is realizable as a Lagrangian extension of Z/2 Z, and moreover, / ` → 1 / ` , / ` / ` that the image of the map Lex(Z 2 Z) ExtZ(Z 2 Z Z 2 Z) is 2Z 2 Z. Thus the / ` → 1 / ` , / ` / cokernel of the map Lex(Z 2 Z) ExtZ(Z 2 Z Z 2 Z) must be isomorphic to Z 2Z. 125

Lemma 4.4.1.7 ([13], Lemma 3.16). The functor C is isomorphic to the functor

B 7→ B2 = {b ∈ B | 2b = 0}.

Proof. φ 1 B, B τ Since the image of coincides with ExtZ(b )0, we have a short exact sequence

{0} / Ext1 (B, B)τ / Ext1 (B, B)τ / C(B) / {0} Z b 0 Z b

1 B, B τ → H1 / , B⊗B Thus the homomorphism ExtZ(b ) (Z 2Z ) deﬁned in § 4.3 factors through 1 an embedding C(B) → H (Z/2Z, B⊗B). By remark 4.3.1.2, this is a natural transformation of linear functors in B. To show that it is an isomorphism, it suﬃces to check it for cyclic B. It is obvious for B of odd order. Remark 4.3.1.2 says that

1 ` H (Z/2Z, B⊗B) is isomorphic to Z/2Z. Finally, remark 4.4.1.6 shows that C(Z/2 Z) is 1 also isomorphic to Z/2Z, making the map C(B) → H (Z/2Z, B⊗B) an isomorphism.

Finally, we summarize the results of this section as a theorem.

Theorem 4.4.1.8 ([13], Theorem 3.17). For any ﬁnite Abelian group B, the following sequence is exact:

φ { } −→ B/ B −→ B −→ 1 B, B τ −→ B −→ { } 0 2 Lex( ) ExtZ(b ) 2 0

Proof. Follows directly from corollary 4.4.1.5 and lemma 4.4.1.7.

4.5 Explicit Computation of the Homomorphism Lex(L) → H3(bL, k∗)

Here, we describe the homomorphism Lex(L) → H3(A/L, k∗) explicitly by giving a 3-cocycle β ∈ Z3(A/L, k∗) representing the Lagrangian extension (A, q, ι). Upon identiﬁcation of A/L with bL, this gives a formula for the associator β of the Lagrangian algebra corresponding to L. 126

4.5.1 Lagrangian Algebras in Pointed Categories

Let now C be a braided monoidal category with the braiding cX,Y : X ⊗ Y → Y ⊗ X.

65 Recall from [40] that the category CR of right R-modules is monoidal with respect to the

loc relative tensor product ⊗R, and that the category CR of local modules is braided. Computations will be greatly simpliﬁed by taking a strict model of the category C(A, α, c). Simple objects of C(A, α, c) are labeled by elements of A. Passing to a strict model gives rise to fusion isomorphisms ιa,b : I(a) ⊗ I(b) → I(a + b) for a, b ∈ A. Let a, b, c ∈ A. By going around the diagram

ιa,b⊗Id I(a) ⊗ I(b) ⊗ I(c) / I(a + b) ⊗ I(c)

Id⊗ιb,c ιa+b,c I(a) ⊗ I(b + c) / I(a + b + c) ιa,b+c

clockwise, we obtain an automorphism of the simple object I (a + b + c) ∈ C(A, α, c) that

is precisely α(a, b, c) · IdI(a+b+c). The braiding in C(A, α, c) is given by (1.12). Recall that the category C(A, α, c) can also be written as C(A, q), where q ∈ Q(A, k∗)

3 ∗ is the unique quadratic form corresponding to the class of the pair (α, c) ∈ Hab(A, k ). Now let B ≤ A be isotropic with respect to q, and let R(B) ∈ C(A, q) be the object given by

M R(B) = I(b). (4.22) b∈B

2 ∗ Let η ∈ C (B, k ) be such that ∂η = α|B, and deﬁne the multiplication map µ : R(B) ⊗ R(B) → R(B) by

0 0 µ(b, b ) = η(b, b ) · ιb,b0 (4.23)

The coboundary condition on η makes this multiplication associative. This makes R(B) into an associative algebra in C(A, q). R(B) is commutative if and only if

0 0 0 0 η(b, b ) · ιb,b0 = c(b, b )η(b , b) · ιb0,b for all b, b ∈ B.

65 In the case where C has coequalizers. 127

The natural forgetful functor C(A, q) → V(A, α) (forgetting the quadratic structure) is

a tensor equivalence. This induces a tensor equivalence C(A, q)R(B) → V(A, α)R(B). Let

p : A → A/B be the canonical projection. The associated direct image functor p∗ takes

3 ∗ V(A, α) to V(A/B, β) for some β ∈ H (A/B, k ), and its restriction to V(A, α)R(B) is a

tensor equivalence V(A, α)R(B) 'V(A/B, β). Thus the composition

forget p∗ C(A, q)R(B) / V(A, α)R(B) / V(A/B, β)

exhibits a tensor equivalence between C(A, q)R(B) and V(A/B, β). Recall that Lagrangian algebras in a pointed nondegenerate braided fusion category

C(A, q) correspond to Lagrangian subgroups of A. Let L A be a Lagrangian subgroup, and let R(L) ∈ C(A, q) be the corresponding Lagrangian algebra. Let

J : C(A, q) → C(A, q)R(L) be the free module functor deﬁned by J(X) = X ⊗ R(L), with

corresponding right R(L)-multiplication given by νX : J(X)⊗R(L)R(L) → J(X). The functor

J gives rise to a natural collection of isomorphisms JX,Y : J(X) ⊗R(L) J(Y) → J(X ⊗ Y) induced from the composition

IdX⊗cR(L),Y ⊗IdR(L) IdX⊗IdY ⊗µ X ⊗ R(L) ⊗ Y ⊗ R(L) / X⊗Y⊗R(L)⊗R(L) / X⊗Y⊗R(L) .

The maps JX,Y are R(L)-balanced in the sense that they make the diagram

IdJ(X)⊗cR(L),J(Y) J(X) ⊗R(L) R(L) ⊗R(L) J(Y) / J(X) ⊗R(L) J(Y) ⊗R(L) R(L)

α IdJ(X)⊗νY J(X) ⊗R(L) R(L) ⊗R(L) J(Y) J(X) ⊗R(L) J(Y)

νX⊗IdJ(Y) JX,Y J(X) ⊗R(L) J(Y) / J(X⊗Y) JX,Y 128

commute for all X, Y ∈ C(A, q), and coherent in the sense that they make the diagram

JX,Y ⊗IdJ(Z) J(X)⊗R(L) J(Y)⊗R(L) J(Z) / J(X ⊗ Y)⊗R(L) J(Z)

IdJ(X)⊗JY,Z JX⊗Y,Z

J(X)⊗R(L) J(Y⊗Z) / J(X⊗Y⊗Z) JX,Y⊗Z

∼ commute for all X, Y, Z ∈ C(A, q). They also give rise to an isomorphism J(I) −→ R(L) according to the diagram

∼ J(I) / R(L) :

ρR(L) I⊗R(L)

Here, I ∈ C(A, q) denotes the monoidal unit and ρR(L) denotes the right unit isomorphism in C(A, q).

We will also take a strict model of the category of modules C(A, q)R(L). All simple R(L)-modules are induced from simple objects of C(A, q). For a ∈ A, denote J(I(a)) by J(a), and deﬁne maps of R(L)-modules

φa,b : J(a) ⊗R(L) J(b) → J(a + b) by the following diagram:

φa,b J(a) ⊗R(L) J(b) / J(a + b)

J(I(a))⊗R(L) J(I(b)) J(I(a + b)) 7

JI(a),I(b) ) J(ιa,b) J(I(a)⊗I(b)) 129

For a, b, c ∈ A, the clockwise composition of the diagram

φa,b⊗IdJ(c) J(a)⊗R(L) J(b)⊗R(L) J(c) / J(a + b)⊗R(L) J(c) (4.24)

IdJ(a)⊗φb,c φa+b,c

J(a)⊗R(L) J(b + c) / J(a + b + c) φa,b+c

is precisely α(a, b, c) · IdJ(a+b+c).

For ` ∈ L, deﬁne θ` : I(`) ⊗ R(L) → R(L) by

M 0 θ` = η `, ` · ι`,`0 . `0∈L

This gives an isomorphism θ` : J(`) → J(0) of right R(L)-modules in C(A, q). The collection of isomorphisms θ` allows one to deﬁne, for any a ∈ A and any ` ∈ L,

isomorphisms of R(L)-modules ϑa,` : J(a + `) → J(a) by the diagram

ϑa,` J(a + `) / J(a) O

φa,`

J(a)⊗R(L) J(`) / J(a)⊗R(L)R(L) IdJ(a)⊗θ`

0 Now deﬁne ϑ`,a : J(` + a) → J(a) by the diagram

0 ϑ`,a J(` + a) / J(a) O

φ`,a

J(`)⊗R(L) J(a) / R(L)⊗R(L) J(a) θ`⊗IdJ(a)

0 Commutativity of addition implies that the maps ϑa,` and ϑ`,a diﬀer by a nonzero scalar, 0 viz., ϑ`,a = c(a, `) · ϑa,`. The collections ϑ establishes that up to isomorphism, simple R(L)-modules are labeled by elements of the quotient group A/L. 130

To finally define a strict model of the category C(A, q)R(L), choose a set-theoretic section s : A/L → A of the canonical projection. Define γ : A/L × A/L → L by

γ (x, y) = s(x) + s(y) − s (x + y), and deﬁne Jx,y : J(s(x))⊗R(L) J(s(y)) → J(s(x + y)) as the composition

φs(x),s(y) ϑs(x+y),γ(x,y) J(s(x))⊗R(L) J(s(y)) / J(s(x) + s(y)) / J(s(x + y)) .

The tensor category C(A, q)R(L) of right R(L)-modules is equivalent to V(A/L, β) for some β ∈ Z3(A/L, k∗). The clockwise composition of the arrows of the diagram

Jx,y⊗IdJ(s(z)) J(s(x))⊗R(L) J(s(y))⊗R(L) J(s(z)) / J(s(x) + s(y))⊗R(L) J(z) (4.25)

IdJ(s(x))⊗Jy,z Jx+y,z

J(s(x))⊗R(L) J(s(y + z)) / J(s(x + y + z)) Jx,y+z

is precisely β(x, y, z) · IdJ(s(x+y+z)), and so we need only compute this composition to see the value of the associator β(x, y, z). We now expand the diagram (4.25). In doing so, we will suppress the tensor product symbols for the sake of compactness of notation. We will also suppress labels on identity morphisms. Juxtaposition of objects in the following diagram should be understood to denote their tensor product over R(L). Moreover, we will write R instead of R(L) where applicable. 131

φs(x),s(y)Id ϑs(x+y),γ(x,y) J(s(x))J(s(y))J(s(z)) / J(s(x) + s(y))J(s(z)) / J(s(x + y))J(s(z))

Idφs(y),s(z) φs(x)+s(y),s(z) φs(x+y),s(z)

φs(x),s(y)+s(z) ϑs(x+y)+s(z),γ(x,y) J(s(x))J(s(y) + s(z)) / J(s(x) + s(y) + s(z)) / J(s(x + y) + s(z))

Idϑs(y+z),γ(y,z) ϑs(x)+s(y+z),γ(y,z) ϑs(x+y+z),γ(x+y,z)

J(s(x))J(s(y + z)) / J(s(x) + s(y + z)) / J(s(x + y + z)) φs(x),s(y+z) ϑs(x+y+z),γ(x,y+z) (4.26) It is straightforward to see that the clockwise reading of the upper left cell of the diagram (4.26) contributes a factor of α(s(x), s(y), s(z)).

Lemma 4.5.1.1 ([13], Lemma A.1). The lower left cell of the diagram (4.26) contributes a factor of α(s(x), s(y + z), γ(y, z))−1.

Proof. Let a = s(x), b = s(y + z), ` = γ(y, z). Consider the diagram

φa,b+` J(a)J(b + `) / J(a + b + `) h 6 Idφb,`

φa+b,` φa,bId J(a)J(b)J(`) / J(a + b)J(`)

Idϑb,` IdIdθ` Idθ` ϑa+b,` J(a)J(b)R / J(a + b)R φa,bId

J(a)J(b) / J(a + b) φa,b

The right cell has the form

φa+b,` J(a + b)J(`) / J(a + b + `)

Id⊗θ` ϑa+b,` J(a + b)R J(a + b) 132

and commutes on the nose, as it is the deﬁnition of ϑa+b,`. The bottom cell has the form

φa,b⊗Id J(a)J(b)R / J(a + b)R

J(a)J(b) / J(a + b) φa,b

and commutes for tautological reasons. The left cell has the form

Id⊗φb,` J(a)J(b + `) o J(a)J(b)J(`)

Id⊗ϑb,` Id⊗Id⊗θ` J(a)J(b) J(a)J(b)R

J(a) plays no part. This cell is the deﬁnition of IdJ(a)⊗ϑb,` with orientation reversed.

−1 Hence it deﬁnes ϑb,` and thus commutes on the nose. The center cell has the form

φa,b⊗Id J(a)J(b)J(`) / J(a + b)J(`)

Id⊗Id⊗θ` Id⊗θ` J(a)J(b)R / J(a + b)R φa,b⊗Id

−1 −1 This cell commutes because its composition gives the scalar φa,bθ` φa,bθ`, and scalars commute pairwise. The top cell is the only cell that does not commute on the nose. It has the form

φa,b+` J(a)J(b + `) / J(a + b + `) O O Id⊗φb,` φa+b,`

J(a)J(b)J(`) / J(a + b)J(`) φa,b⊗Id

−1 By (4.24), its clockwise composition gives α(a, b, `) . The result follows.

Lemma 4.5.1.2 ([13], Lemma A.2). The upper right cell of the diagram (4.26) contributes a factor of

α(γ(x, y), s(x + y), s(z))c(s(x + y) + s(z), γ(x, y)) . c(s(x + y), γ(x, y)) 133

Proof. Let now ` = γ(x, y), a = s(x + y), and b = s(z), and consider the diagram

0 ϑ`,aId J(` + a)J(b) / J(a)J(b) h

φ`,aId θ`IdId J(`)J(a)J(b) / RJ(a)J(b)

φ`+a,b Idφa,b Idφa,b φa,b J(`)J(a + b) / RJ(a + b) θ`Id φ`,a+b

v J(` + a + b) 0 / J(a + b) ϑ`,a+b

The rightmost cell has the form

RJ(a)J(b) J(a)J(b)

Id⊗φa,b φa,b RJ(a + b) J(a + b)

and commutes on the nose for tautological reasons. The center cell has the form

θ`⊗Id⊗Id J(`)J(a)J(b) / RJ(a)J(b)

Id⊗φa,b Id⊗φa,b J(`)J(a + b) / RJ(a + b) θ`⊗Id and also commutes on the nose. The leftmost cell has the form

φ`,a⊗Id J(` + a)J(b) o J(`)J(a)J(b)

φ`+a,b Id⊗φa,b J(` + a + b) o J(`)J(a + b) φ`,a+b

and contributes a factor of α(`, a, b). The bottom cell has the form

θ`⊗Id J(`)J(a + b) / RJ(a + b)

φ`,a+b J(` + a + b) 0 / J(a + b) ϑ`,a+b 134

0 and is the deﬁnition of ϑ`,a+b. Similarly, the top cell has the form

0 ϑ`,a⊗Id J(` + a)J(b) / J(a)J(b) O φ`,a⊗Id

J(`)J(a)J(b) / RJ(a)J(b) θ`⊗Id⊗Id

0 0 and is the deﬁnition of ϑ`,a. Now, in the diagram (4.26), ϑ s do not appear. Instead, ϑs appear. Comparing ϑ0 with ϑ gives factors of c(a + b, `) from the bottom cell and c(a, `)−1 from the top cell. The inverse appears in the second factor due to the orientation being reversed relative to the deﬁnition of ϑa,`. The result follows.

Lemma 4.5.1.3 ([13], Lemma A.3). The lower right cell of the diagram (4.26) contributes a factor of η(γ(x, y + z), γ(y, z)) . η(γ(x + y, z), γ(x, y))

Proof. We split the lower right cell of the diagram (4.26) into two triangles as follows:

ϑs(x+y)+s(z),γ(x,y) J(s(x) + s(y) + s(z)) / J(s(x + y) + s(z)) (4.27)

ϑs(x)+s(y+z),γ(y,z) ϑs(x+y+z),γ(x+y,z) ϑs(x+y+z),γ(x,y,z)

& J(s(x) + s(y + z)) / J(s(x + y + z)) ϑs(x+y+z),γ(x,y+z)

Let us now consider the general form of such triangles:

ϑa,`+`0 J(a + ` + `0) / J(a) (4.28) :

ϑ 0 ϑ a+`,` ' a,` J(a + `) 135

Expanded, this becomes

ϑa,`+`0 J(a + ` + `0) / J(a) a k O

φa,`+`0 Idθ`+`0 J(a)J(` + `0) / J(a)R φa+`,`0 O Idφ`,`0

J(a + `)J(`0) o J(a)J(`)J(`0) φa,`Id

ϑa+`,`0 Idθ`0 IdIdθ`0 ϑa,` J(a + `) o J(a)J(`)R J(a)R φa,`Id 8

Idθ` J(a)J(`)

φ a,` J(a + `) s / J(a + `) IdJ(a+`)

All of the quadrilaterals in this diagram commute on the nose, leaving only the right-center cell, which collapses to

Id⊗θ`+`0 J(a)J(` + `0) / J(a)R (4.29) O O Id⊗φ`,`0 Id⊗θ`

J(a)J(`)J(`0) / J(a)J(`) Id⊗Id⊗θ`0

Now, since J(a) plays no part in the cell (4.29), we may omit it and rewrite (4.29) as

θ`+`0 J(` + `0) / R O O φ`,`0 θ`

J(`)J(`0) / J(`) Id⊗θ`0

Going clockwise through this diagram from the bottom left gives us a factor of η`, `0. Therefore, the lower triangle of (4.27) contributes a factor of η(γ(x, y + z), γ(y, z)), and the upper triangle contributes a factor of η(γ(x + y, z), γ(x, y))−1. The inverse appears in the second factor because the orientation of the upper triangle is reversed relative to

(4.28). 136

Finally, we summarize the results of this section as a theorem.

Theorem 4.5.1.4 ([13], Theorem A.4). The category C(A, q)R(L) of right modules over a Lagrangian algebra R(L) ∈ C(A, q) is equivalent, as a monoidal category, to the category V(A/L, β) of A/L-graded vector spaces, where β ∈ Z3(A/L, k∗) is given by the formula

β(x, y, z) = (4.30)

α(s(x), s(y), s(z))α(γ(x, y), s(x + y), s(z))c(s(x + y) + s(z), γ(x, y))η(γ(x, y + z), γ(y, z)) = α(s(x), s(y + z), γ(y, z))c(s(x + y), γ(x, y))η(γ(x + y, z), γ(x, y))

Proof. Follows from lemmata 4.5.1.1, 4.5.1.2, and 4.5.1.3.

4.6 Examples

∗ Example 4.6.1.1. Let A = B⊕bB for an Abelian group B, and let qstd : B⊕bB → k be a

quadratic function given by qstd(b, χ) = χ(b). The subgroup L = B⊕{1} is Lagrangian with respect to q. The character group bL can be identiﬁed with bB. We have a braided equivalence C B⊕bB, qstd 'Z bB .

Example 4.6.1.2. Let A = B⊕B for a nondegenerate quadratic Abelian group (B, q). Equip A with a quadratic function q⊕q−1. The diagonal subgroup ∆ = {(b, b)| b ∈ B} is

Lagrangian. The character group bL can be identiﬁed with Bˆ. We have a braided equivalence

−1 C B⊕B, q⊕q 'Z bB, αq .

2 2 Example 4.6.1.3. Let A = Z/m Z be a cyclic group of order m for an odd m and let 2 ∗ s2 ∗ qε : Z/m Z → k be a quadratic function given by q(s) = ε , where ε ∈ k is a primitive 2 2 root of unity of order m . The subgroup L = mZ/m Z is Lagrangian with respect to qε. The character group bL can be identiﬁed with Z/mZ via

jml Z/mZ → bL, j 7→ χ j, χ j(ml) = ε . 137

2 For ν ∈ k∗ such that νm = 1, denote by

x(y+z−[y+z]m) αν(x, y, z) = ν , (4.31)

where [x]m is the residue of x ∈ Z modulo m (taking values between 0 and m − 1). Note

that αν is a normalised 3-cocycle of Z/mZ and that the assignment ν 7→ αν gives an 3 ∗ th isomorphism between H (Z/mZ, k ) and the group µm(k) of m roots of 1 in k. We have a braided equivalence

2 C Z/m Z, qε 'Z(Z/mZ, αεm ) .

In order that the braided category Z(Z/mZ, α) be pointed, it is necessary that the cocycle 1 ∗ α be a coboundary. Given z ∈ Z/mZ, let cz ∈ C (Z/mZ, k ) be such that

(∂cz)(x, y) = α(x, y|z) for all x, y ∈ Z/mZ. Computing α(x, y|z) for the cocycle α from (4.31) gives

−z x+y− x+y α (x, y|z) = ν ( [ ]m) . (4.32)

Used inductively, the coboundary equation allows one to describe cz. Namely cz is

completely determined by its value at 1 ∈ Z/mZ: x (cz(1)) cz(x) = (4.33) Yx−1 α (i, 1|z) i=1 Indeed, for x ∈ Z/mZ, we have c (x − 1) c (1) c (x) = c ((x − 1) + 1) = z z . z z α (x − 1, 1|z)

Moreover, the formula (4.33) yields a normalized 1-cochain of Z/mZ with coeﬃcients in ∗ k only if cz(m) = 1. This requires that m−1 m Y (cz(1)) = α (i, 1|z). (4.34) i=1 It follows from (4.32) that the only nontrivial value of α(i, 1|z) is α(m − 1, 1|z) = ν−zm. That gives

m −zm (cz(1)) = ν . 138

−s(z) Making a choice cz(1) = ν gives

−s(z)s(x) cz(x) = ν , x ∈ Z/mZ.

Using z 7→ (z, cz) as a section of Pic(Z(Z/mZ, α)) → Z/mZ, we can write down the 2 extension class as an element of Z (Z/mZ, Z\/mZ). Namely, for z, w ∈ Z/mZ, we can deﬁne the character χz,w ∈ Z\/mZ by

α (g|z, w) cz+w(g) χz,w (g) = . (4.35) cz(g)cw(g)

Note that the isomorphism

m k Z\/mZ → Z/mZ, χ 7→ k, such that χ(1) = (ν ) identiﬁes the extension cocycle (4.35) with the canonical extension cocyle

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Complex

It is well known that the standard diﬀerential (1.3) comes from a cosimplicial n+1 66 X i structure on the standard complex ∂ = (−1) ∂i. Here, i=0 n n+1 ∂i : C (G, M) → C (G, M), i = 0, ..., n + 1 are deﬁned by    c(g2, ..., gn+1), i = 0   ∂i(c)(g1, ..., gn+1) =  c(g1, ..., gi−1, gigi+1, gi+2, ..., gn+1), 1 ≤ i ≤ n    c(g1, ..., gn), i = n + 1 and satisfy the cosimplicial identities

∂ j∂i = ∂i∂ j−1, i < j .

The group of automorphisms Aut(G) of the group G acts on the complex C∗(G, M) via

−1 −1 n φ(c)(g1, ..., gn) = c(φ (g1), ..., φ (gn)), φ ∈ Aut(G), c ∈ C (G, M) .

−1 For g ∈ G denote by φg the corresponding inner automorphism φg(x) = gxg .

Proposition A.1.1.1 ([15]). The action of inner automorphisms on the complex C∗(G, M) is homotopically trivial. More precisely, for any c ∈ Cn(G, M) and g ∈ G, we have

g g φg(c) = c + ∂(h (c)) − h (∂(c)), where

n−1 g −1 X i −1 −1 −1 h (c)(g1, ..., gn−1) = c(g , g1, ..., gn−1) + (−1) c(g g1g, ..., g gig, g , gi+1, ..., gn−1) . i=1 (A.1) 66 See, e.g.,[20]. 144

n−1 g X i g Proof. For an n-cochain c, write h (c) = (−1) hi (c), where i=0   c(g−1, g , ..., g ), i = 0 g  1 n−1 hi (c)(g1, ..., gn−1) =   −1 −1 −1  c(g g1g, ..., g gig, g , gi+1, ..., gn−1), 1 ≤ i ≤ n − 1

The following identities

g g h0∂0 = Id, hn∂n+1 = φg,   g  ∂ih j−1, i < j  g  g h ∂i = h ∂ , i j j  i−1 i = , 0   g  ∂i−1h j , i > j + 1 g show that hi is a cosimplicial homotopy between φg and the identity Id. We now prove the identities. h i Assume that i = j = 0. Then ∂0 (c) (g1,..., gn+1) = c (g2,..., gn+1). Call this

h g i −1 function b. Then h0 (b) (g1,..., gn) = b g , g1,..., gn = c (g1,..., gn). The ﬁrst identity follows. Assume that i = n + 1 and j = n. Note that this value of j makes sense, as it entails

acting upon the (n + 1)-cochain ∂n+1(c). We obtain

h i h g i ∂n+1 (c) (g1,..., gn+1) = c (g1,..., gn). Call this function b. Then hn (b) (g1,..., gn) =

−1 −1 −1 −1 −1 −1 −1 b g g1g, g g2g,..., g gng, g = c g g1g, g g2g,..., g gng = φg (c) by deﬁnition of the action of Aut(G) on the complex C∗ (G, M). The second identity follows. The proof of the third identity must be split into several cases. We ﬁrst assume that i < j. If i = 0 and

h n i j = 1, then we have ∂0 (c) (g1,..., gn+1) = c (g2,..., gn+1). Here, the superscript on ∂ denotes the degree of the diﬀerential map under consideration. Call this function b. Then

h g i −1 −1 −1 h1 (b) (g1,..., gn) = b g g1g, g , g2,..., gn = c g , g2,..., gn . On the other h g i −1 hand, h0 (c) (g1,..., gn−1) = c g , g1,..., gn−1 . Call this function b. Then h n−1 i −1 ∂0 (b) (g1,..., gn) = b (g2,..., gn) = c g , g2,..., gn , and the desired equality follows. 145

h n i Now assume that i = 0 and j ≥ 2. As above, ∂0 (c) (g1,..., gn+1) = c (g2,..., gn+1). Call this function b. Then

h g i −1 −1 −1 −1 h j (b) (g1,..., gn) = b g g1g, g g2g,..., g g jg, g , g j+1,..., gn = −1 −1 −1 = c g g2g,..., g g jg, g , g j+1,..., gn . On the other hand,

h g i −1 −1 −1 h j−1 (c) (g1,..., gn−1) = c g g1g,..., g g j−1g, g , g j,..., gn−1 . Call this function h n−1 i b. Then ∂0 (b) (g1,..., gn) = g1.b (g2,..., gn) = −1 −1 −1 −1 = g g1g .c g g2g,..., g g jg, g , g j+1,..., gn , and the desired equality follows. Finally, assume that i ≥ 1 and j ≥ 2. Note that in this case, it is not possible to have

h n i i = n + 1. We have ∂i (c) (g1,..., gn+1) = c (g1,..., gi−1, gigi+1, gi+2,..., gn+1). Call this h g i function b. Then we have h j (b) (g1,..., gn) = −1 −1 −1 −1 = b g g1g, g g2g,..., g g jg, g , g j−1,..., gn =

−1 −1 −1 −1 = c g g1g,..., g gigi+1g,..., g g jg, g , g j+1,..., gn . On the other hand,

h g i −1 −1 −1 h j−1(c) (g1,..., gn−1) = c g g1g,..., g g j−1g, g , g j,..., gn . Call this function b. h n−1 i Then ∂i (b) (g1,..., gn) = b (g1,..., gi−1, gigi+1, gi+2,..., gn) =

−1 −1 −1 −1 = c g g1g,..., g gigi+1g,..., g g jg, g , g j+1 ..., gn , and the desired equality follows.

We next assume that i = j , 0. We ﬁrst consider the case where i = j = 1.

h n i ∂1(c) (g1,..., gn+1) = c (g1g2, g3,..., gn+1). Call this function b. Then h g i −1 −1 −1 −1 h1(b) (g1,..., gn) = b g g1g, g , g2,..., gn = c g g1gg , g2,..., gn ..., gn = −1 = c g g1, g2,..., gn . On the other hand,

h g i −1 −1 h0(b) (g1,..., gn) = b g , g1,..., gn = c g g1, g2,..., gn , and the desired equality follows. We now consider the case where i = j ≥ 2. We have

h n i ∂i (c) (g1,..., gn+1) = c (g1,..., gi−1, gigi+1, gi+2,..., gn+1). Call this function b. Then

h g i −1 −1 −1 hi (b) (g1,..., gn) = b g g1g,..., g gig, g , gi+1,..., gn = 146

−1 −1 −1 −1 −1 −1 = c g g1g,..., g gigg , gi+1,..., gn = c g g1g,..., g gi−1g, g gi, gi+1,..., gn .

h g i −1 −1 −1 On the other hand, hi−1(b) (g1,..., gn) = b g g1g,..., g gi−1g, g , gi,..., gn = −1 −1 −1 = c g g1g,..., g gi−1g, g gi, gi+1,..., gn , and the desired equality follows. Finally, we assume that i > j + 1. We ﬁrst assume that j = 0 and i ≥ 2. As in the

h n i immediately preceding case, ∂i (c) (g1,..., gn+1) = c (g1,..., gi−1, gigi+1, gi+2,..., gn+1).

h g i −1 Call this function b. Then h0(b) (g1,..., gn) = b g , g1,..., gn = −1 h g i = c g , g1,..., gi−2, gi−1gi, gi+1,..., gn . On the other hand, h0(c) (g1,..., gn−1) = −1 h n−1 i = c g , g1,..., gn−1 . Call this function b. Then ∂i−1 (b) (g1,..., gn) = −1 = b (g1,..., gi−2, gi−1gi, gi+1,..., gn) = c g , g1,..., gi−2, gi−1gi, gi+1,..., gn , and the desired equality follows. Lastly, we consider the case where j > 0. We have h i ∂i(c) (g1,..., gn+1) = c (g1,..., gi−1, gigi+1, gi+1,..., gn). Call this function b. Then

h g i −1 −1 −1 h j (b) (g1,..., gn) = b g g1g,..., g g jg, g , g j+1,..., gn = −1 −1 −1 = c g g1g,..., g g jg, g , g j+1,..., gi−1gi, gi+1,..., gn . On the other hand,

h g i −1 −1 −1 h j (c) (g1,..., gn−1) = c g g1g,..., g g jg, g , g j+1,..., gn−1 . Call this function b. h n−1 i Then ∂i−1 (b) (g1,..., gn) = b (g1,..., gi−2, gi−1gi, gi+1,..., gn) = −1 −1 −1 = c g g1g,..., g g jg, g , g j+1,..., gi−1gi, gi+1,..., gn , and the proof of the identities is completed. With the identities proven, we now turn our attention to the proof of the result. We begin by writing n−1 n+1 g X i g X j h = (−1) hi , ∂ = (−1) ∂ j. i=0 j=0 g g We claim that φg − Id = ∂h − h ∂. Indeed, we have

 n   n−1   n   n+1  X  X g X g X  ∂hg − hg∂ =  (−1) j∂   (−1)i h  −  (−1)ih   (−1) j∂  =  j  i   i   j j=0 i=0 i=0 j=0

n n−1 n n+1 X X i+ j g X X i+ j g = (−1) ∂ jhi − (−1) h j ∂i = i=0 j=0 j=0 i=0 147

X i+ j g X i+ j g X i+ j g X i+ j g = (−1) ∂ih j + (−1) ∂ jhi − (−1) h j ∂i − (−1) h j ∂i+ 0

X i+ j g X i+ j g g g + (−1) h j ∂i − (−1) h j ∂i − h0∂0 + hn∂n+1 1≤i≤n 0≤ j

and the proof is completed.

Remark A.1.1.2. It follows from proposition A.1.1.1 that the group of inner

∗ automorphisms Inn(G) = {φg| g ∈ G} acts trivially on the cohomology H (G, M). Therefore the action of Aut(G) on C∗(G, M) gives rise to an action of the outer automorphism group Out(G) = Aut(G)/Inn(G) on H∗(G, M). ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

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