The Brauer Group of a Braided Monoidal Category

JOURNAL OF ALGEBRA 202, 96᎐128Ž. 1998 ARTICLE NO. JA977295

Fred Van Oystaeyen and Yinhuo Zhang

Department of Mathematics, Uni¨ersity of Antwerp() UIA , B-2610 Belgium

Communicated by Michel Van den Bergh View metadata, citation and similar papers at core.ac.uk brought to you by CORE Received August 1, 1996 provided by Elsevier - Publisher Connector

INTRODUCTION

Representation theory of finite groups prompted the introduction of the Brauer group of a field and also of its Schur subgroups. The theory of quadratic forms and spaces introduced the ޚr2ޚ-graded variant of the Brauer group together with the typical Clifford algebras and related representations; cf. the work of Wallwx 29 . The extension of the ޚr2ޚ- graded theory to gradings by other cyclic groups and consequently to abelian groups, replacing the usual Clifford algebras by the so-called generalized Clifford algebras, is then natural from the algebraic point of view. However, extension to nonabelian groups did not seem to be possi- ble, at least not by simple modifications of the abelian theory. At the same time, the generalization of the Brauer group of graded algebras obtained by Longwx 12, 13 made use of Hopf algebras but againŽ. co- commutativity conditions were present. The recent interest in quantum groups motivated the authors to introduce the Brauer group of a quantum group as the Brauer group of crossed module algebrasŽ also called quantum Yang᎐ Baxter module algebras. inwx 4, 5 . The use of the category of crossed modules originated in the case of group algebras; cf. Whiteheadwx 30 . Module algebrasŽ. or coalgebras of this crossed type were introduced by Radford inwx 24 and used by Majid in the description of modules over Drinfel’d quantum doubles. The non-Ž. co- commutativity aspects of the theory concerning the Brauer group in terms of crossed module algebras do not have a counterpart in classical theory. Note that a first unifying categorical theory has been obtained by Pareigis inwx 23 ; this theory deals with the Brauer group of a

0021-8693r98 $25.00 Copyright ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. BRAIDED MONOIDAL CATEGORY 97 symmetric monoidal category. The restriction to a symmetric monoidal category, probably viewed as just a commutativity condition, does in fact restrict attention to trivial actions and coactions. In order to formulate the construction of the Brauer group of a quantum group into a categorical framework, one has to extend the categorical construction to the situation of braided monoidal categories, recently considered by Majidwx 15, 18 and Joyal and Street wx 9 . Indeed, the fact that the category of crossed modules is a braided monoidal category is at the heart ofwx 4, 5 so the consideration of the general braided case seems to be a natural extension. The Brauer group of a braided monoidal category generalizes precisely the notion of the Brauer᎐Long group to the categorical setting. This is most obvious when considering Hopf algebras over ރ. It is clear that the Brauer group of such a Hopf algebra is still large and may even fail to be an abelian torsion groupwx 28 , where the usual Brauer group of ރ is trivial. The extra complexity apparent in the Brauer group of a Hopf algebra or a braided monoidal category is entirely created by the Hopf algebra action and coaction properties, resp. by the braiding properties. Concrete calculations may therefore be linked to problems of a more representation theoretic nature! In Section 1, we recall some definitions concerning braided monoidal categories and the objects in them. The elementary algebras in a braided monoidal category C are studied in Section 2. Those algebras are still closely related to the Morita theory of C as defined by Pareigiswx 23 . This allows us to characterize Azumaya algebras in C and leads to the definition of the Brauer group of C which need not be an abelian group. The examples in Section 3 establish how all known Brauer groups derive from this unifying definition, for example, the Brauer group of a field, the Brauer group of a scheme, the Brauer᎐Wall groupŽ. see Example 3.9 , the graded Brauer groupŽ. see Example 3.8 , the Brauer group with respect to a bicharacter as defined by Childs, Garfinkel, and Orzech, the Brauer᎐Long groupŽ. see Example 3.11 , and the Brauer group of a quantum group Ž see Example 3.12. . Moreover, a general Rosenberg᎐Zelinsky exact sequence of an Azumaya algebra is presentedŽ. see Proposition 3.14 . In Section 4, we construct a second Brauer group BrXŽ.C of separable algebras in a braided monoidal category C. The main result of this section is Theorem 4.9. The approach to Theorem 4.9 is different from that of Pareigis even in the symmetric case where more conditions are required Ž.see Examples 4.13 . In order to stay close to the classic cases, one then has to demand that the braided monoidal category considered is cocomplete abelian and has a nice unit. There is a functorial monomorphism from the second Brauer group BrXŽ.C to the Ž first . Brauer group BrŽ.C of a braided monoidal category C. The monomorphism becomes an isomorphism if the unit of C is a projective object in C. 98 VAN OYSTAEYEN AND ZHANG

In order to know how far the Brauer group of a braided monoidal category is from the Brauer group of a quantum group, we have to study the behavior of the Brauer group of a braided monoidal category under base change. The Tannaka᎐Krein theorem will play an important role hereŽ. see Theorem 5.3 .

1. PRELIMINARIES

Let us briefly recall the definition of a braided monoidal category. For full detail we refer towx 9, 18 . A braided monoidal category is Ž.C, m, I, ␾ where Ž.C, m, I is a monoidal category satisfying the coherence conditions wx14 , and ␾, called a braiding, is a natural transformation between the two op functors m and m Ž.with opposite product from C = C ª C. This is a collection of functorial isomorphisms ␾ V, W : V m W ª W m V obeying two ‘‘hexagon’’ coherence identities. In our notation these are

␾ ␾ ␾ ␾ ␾ ␾ VmW,ZVs ,ZW( ,ZV, ,WmZVs ,ZV( ,W.1Ž.

The associativity morphism V m Ž.Ž.W m Z ( V m W m Z is suppressed inŽ. 1 . One easily deduces identities of the following type:

␾ V,IVIs id s ␾ ,V.2Ž.

2 Then morphisms associated with I are suppressed inŽ. 2 . If ␾ s id, then one of the hexagons is superfluous and we have an ordinary symmetric monoidal category. Let C be a braided monoidal category. If P g C, denote by PXŽ.the set CŽ.X, P for X g C. If the functor C Žym P, Q .is representable, the representing object is denoted by wxP, Q . If, for any object Q g C, wxP, Q exists, then the tensor functor ym P has a right adjoint functor wxP, ᎐ , that is, ᭙X, P g C,

CŽ.XmP,Q(CŽ.X,wxP,Q.

Similarly, if the functor CŽ.P m ᎐, Q is representable, then the representing object is denoted by Ä4P, Q . Now the evaluation map

wxP,QXŽ.=PY Ž.ªQX ŽmY ., Žf,p .¬fp²:, is induced by the composition of morphisms; here wxP, Q operates on P from the left side. We say that P is a finite object of C if wxwxP, P and P, I exist and the canonical morphism

PmwxwxP,IªP,P BRAIDED MONOIDAL CATEGORY 99 induced by

PXŽ.mPY Ž.mwxP,IZ Ž.ªPX ŽmYmZ . is an isomorphism. This is equivalent to the existence of a ‘‘dual basis’’ p00mfgPmwxP,IIŽ.such that pf00 ² p :spfor all p g PX Ž .and XgC. Moreover, P is called faithfully projecti¨e if the morphism

wxP,ImwP,PxPªI, induced by the evaluation, is an isomorphism. That is to say, there exists Ž .Ž . ² : Ž . an element f1mwP,Px p1g wxP, I mwP,Px PIwith fp11sid g II.P is said to be a progenerator if P is finite and there is an element f11m p g ŽwxP, I m PI.Ž .such that fp11 ² :sid g II Ž ..If Iis projective in C, then P is faithfully projective if and only if P is a progenerator; cf. wx22 . A monoidal category C is rigid if every object V g C is finite. Denote U U by V the object wxP, I and by evV the evaluation map: V m V ª I. Write coevV for the composite map of the following canonical maps:

U IªwxP,PªVmV.3Ž. Then

coev 6 UUev 6 VVŽ.mVmV,VmŽ.VmVV,4 Ž.

UUcoev 6 UUUUev 6 VVŽ.mVmV,VmŽ.VmVV Ž.5

U compose to idVV and id U , respectively. V is called the left dual of V. The model is that of a finite-dimensional vector spaceŽ or a finitely generated projective module over a commutative ring. .

A functor F from braided monoidal category Ž.ŽC, m, I, ␾C to D, m, J, ␾D . is called a monoidal functor if the natural transformations

␦ : FXŽ.mFY Ž.ªFX ŽmY ., ␰:JªFI are isomorphisms, and ␦, ␰ should be compatible with the associativity morphismsŽ seewx 14, 23 for detail. . If, in addition, F respects the braiding, then F is called a tensor functor. An object A in a monoidal category C is called an algebra if there are two morphisms in C:

␲ : A m A ª A, ␮: I ª A satisfying the associativity and unitary conditions of usual algebras but expressed in diagrams. We recall fromwx 23 some notions concerning modules in a monoidal category. Let C be a monoidal category, and let A, 100 VAN OYSTAEYEN AND ZHANG

B be algebras in C. An object M in C is called an A᎐B-bimodule if there are morphisms in C:

mmAB6 6 AmMM,MmBM satisfying the coherence conditions:

idmmmAB6 mid 6 AmAmMAmMMmBmBMmB

␲midmmAB idm␲

6 6 6 6

mmAB6 6 AmMMMmBM and mmBAŽ.Ž.mid Bs m AAid m m Bon A m M m B. The bimodule category ABC consists of objects in C which have an A᎐B-bimodule structure and morphisms in C which are A᎐B-bilinear. Write ABC, C for the categories AIC and IC B, respectively. Let ABP be an A᎐B-bimodule in C. Then P induces a functor

F : CABª C , X ¬ X mABP g C .

If F is representable, we denote by wxP, ᎐ B the representing object. We say that PBBis faithfully projecti¨e if wxP, B and wxP, P exist and

P P,B P,P, P,B PB mBwxwxª B wxmwP,PxB ª are isomorphisms. Then we have that F is an equivalence if and only if PB is faithfully projective and wxP, P B ( A in C. Let A be an algebra in C. The opposite algebra A of A is defined as follows: A s A as an object in C, but with multiplication ␲ being ␲ (␾. One may easily prove that A is indeed an algebra in C.If A,Bare algebras in C, then A m B with multiplication given by

idm␾mid 6 Ž.Ž.Ž.Ž.A m B m A m BAmAmBmBªAmB is an algebra in C; cf.wx 18 . Denote the preceding algebra by A࠻B.In e e particular, we write A and A for C-en¨eloping algebras A࠻A and A࠻A, respectively. For more details of algebras and Hopf algebras in a braided category, we refer towx 18, 19 . In the sequel, for the sake of simplification, we will often use braiding figures in the proofs. A good and detailed explanation of the diagrammatic methods can be found inwx 16 . For BRAIDED MONOIDAL CATEGORY 101 examples:

PROPOSITION 1.1. Let A, B be algebras in C. A࠻B( B࠻A. Proof. The morphism from B࠻A to A࠻B is defined as the braiding ␾. It is sufficient to check that ␾ is an algebra morphism. That is, to show that

But this is equivalent to showing that ␾␾Ž.m␾Ž.Ž.Ž.id m ␾ m id s id m ␾ m id ␾␾m␾. Note that

␾12,34 s Ž.id m ␾2,3 m id Ž.␾1,2m ␾ 3,4 Ž.id m ␾2,3 m id . Now we have

Ž.Ž.id m ␾2,3 m id ␾␾12,34 1,2m␾ 3,4

s Ž.id m ␾2,3 m id Ž.␾1,2m ␾␾ 3,4 12,34

s Ž.id m ␾2,3 m id Ž.␾1,2m ␾ 3,4 102 VAN OYSTAEYEN AND ZHANG

=Ž.id m ␾2,3 m id Ž.␾1,2m ␾ 3,4 Ž.id m ␾2,3 m id

s␾␾12,34Ž. 1,2m␾ 3,4 Ž.id m ␾2,3 m id , where the subscriptions indicate the positions of the objects.

Recall fromwx 23 that an object M in ABC is called B-coflat if, for all algebras C g C and objects X gBCC , M m BX exists and if the natural morphism M mBBAŽ.Ž.X m Y ª M m X m Y in CDis an isomorphism for N g CD. M is called bicoflat if M is A-coflat and B-coflat.

2. THE ELEMENTARY ALGEBRAS IN C

Let Ž.C, m, I, ␾ be a braided monoidal category. A Morita context in C is a sextuple Ä4A, B, ABBP , Q A, f, g consisting of algebras A, B g C,an A᎐B-bimodule P gABC ,a B᎐A-bimodule Q gBAC , and bilinear morphisms

f : P mBAQ ª A, g : Q m P ª B making commutative the following diagrams:

fmid 6 gmid 6 P mBAQ m PAmAPQm ABPmQBmBQ

idmg ;;idmf

6 6 6 6

;;6 6 PmBABPQmAQ.

A morphism in C, h: M ª N, is said to be rationally surjective if h: MIŽ.ªNI Ž.is surjective. The following result concerning a Morita context can be found inwx 23, Theorems 5.1 and 5.3 and holds in general in a monoidal category.

THEOREM 2.1. LetÄ4 A, B, ABBP , Q A, f, g be a Morita context in C. If P,Q are bicoflat and f, g are rationally surjecti¨eŽ. equi¨alently isomorphisms , then

Ž.1A(wxP,PBB(Ä4Q,Q and B ( wxQ, Q AA( Ä4P, P as algebras in C.

Ž.2P,wxQ,AAB,Ä4Q,B and Q , wxP, B BA, Ä4P, A as bimodules in C.

Ž.3 ABBP,P, Q,and Q A are faithfully projecti¨e.

Proof. The pair of functors Ž.Q mAB᎐, P m ᎐ defines an equivalence between the categories ABC and C. So the isomorphisms concerning objects Ä4᎐ follow fromwx 23, Theorem 5.3 . Shifting left modules to right BRAIDED MONOIDAL CATEGORY 103

modules, we have the equivalence between CABand C defined by the pair of functors Ž.ymABP, ym Q . An argument similar towx 23, Theorem 5.3 yields the other isomorphisms concerning objects wx᎐ .Ž. 3 follows from the definition of faithfully projective objects.

COROLLARY 2.2. Let P be a faithfully projecti¨e object in C. Then UU Ž.1wxP,P(ÄP,P4 as algebras in C. UU Ž.2Ä4P,P(wP,Px as algebras in C. Ž.3wxP,P(Ä4P,P as algebras in C. U Proof. Let A s wxP, P . Then P is an A᎐I-bimodule, and P s wxP, I is an I᎐A-bimodule in C. Let f be the canonical map P m wxwxP, I ª P, P Ä and g be the evaluation map wxP, I mw P, P x P ª I. Then wxP, P , I, P, PU, f, g4 is a Morita context in C. Applying Theorem 2.1, we obtain the algebra isomorphismsŽ. 1 and Ž. 2 . Now we view P as an I᎐A-bimodule via

␾ P m A ª A m P ª P.

Similarly, PU is an A᎐I-bimodule in C. This yields a Morita context in C:

U ½5A, I, AIP , IAP , f , g ,

UU where f: P mA P ª I induced by g and g: P m P ª A induced by f are UU isomorphisms. It follows from Theorem 2.1 that A ( Ä4P, P ( wxP , P .

PROPOSITION 2.3. Let M, N be faithfully projecti¨e objects in C. Then Ž.1wxwxwM,M࠻N,N(MmN,MmN x. Ž.2Ä4Ä4ÄM,M࠻N,N(MmN,MmN 4. Proof. Ž.1 Let X be a finite object in C. We may identify wxX, X with U U X m X via the canonical map f in Corollary 2.2. In fact, X m X is an algebra and f is an algebra isomorphism. The multiplication map of U X m X is

UUidmgmid 6 U Ž.Ž.XmXmXmXXmX,

where g is the evaluation map. Note that

U UUU Ž.MmNswxwxwxMmN,I,M,N,IsM,N,NmM. 104 VAN OYSTAEYEN AND ZHANG

Define a morphism

id ␾ UU UUm ŽNmN.,M6 UU ␪:MmNmNmMMmMmNmN. Clearly ␪ is an isomorphism, and hence the composition of morphisms

UU ⌰:wxMmN,MmN(Ž.Ž.MmNmNmM

␪ UU ªŽ.Ž.MmMmNmN(wxwxM,M࠻N,N is an isomorphism, too. To show that ⌰ is an algebra morphism, it is sufficient to check that ␪ is an algebra morphism. This can be shown by the factorial property of the evaluation map ev. That is,

UU Ž.2 By Corollary 2.2, we have isomorphisms Ä4M, M ( wM , M x UU and Ä4N, N ( wN , N x. Now, applyingŽ. 1 , we obtain the isomorphism Ž. 2 .

PROPOSITION 2.4. Let A be an algebra in C, M a faithfully projecti¨e object in C.

Ž.1A࠻wxwxM,M(M,M࠻A. Ž.2A࠻Ä4Ä4M,M(M,M࠻A. BRAIDED MONOIDAL CATEGORY 105

Proof. We establish statementŽ. 1 . Then statement Ž. 2 follows from Ž. 1 U andŽ. 2 of Corollary 2.2. Identifying wxM, M with M m M , we define a morphism ␩ as the composition of the following morphisms:

y1 UUU␾ mid 6 idm␾ 6 AmMmMMmAmMMmMmA.

It is clear that ␩ is an isomorphism. We show that ␩ is an algebra morphism diagrammatically as follows:

In the last equality, we used the functoriality of the braiding. That is, the multiplication map commutes with the braiding.

3. AZUMAYA ALGEBRAS IN C

In this section we define Azumaya algebras in a braided monoidal category C, and construct the Brauer group of C. Let A be an algebra in C, and Ae, eA the C-enveloping algebras of A. Then A is an object in the category AAe C as well as in C e via

idm␾ 6 ␲ A࠻A m AAmAmAªA and

␾mid 6 ␲ A m A࠻AAmAmAªA. 106 VAN OYSTAEYEN AND ZHANG

THEOREM 3.1. Let C be a braided monoidal category, A an algebra in C. The following statements are equi¨alent: Ž.1 The following two functors are equi¨alence functors:

C ªA࠻A C , X ¬ A m X, Ž.6 CªCA࠻A, X¬XmA.

Ž.2 A is faithfully projecti¨einC,and the following canonical morphisms are isomorphisms: Ž.iF:A࠻AX Ž .ªwxA,AXŽ.,Fa Ž࠻bd .²:sa␲ Žbmd .,where X,YgC,dgAYŽ.,and a␲ Žb m d .g AX ŽmY .. Ž.ii G: A࠻AX Ž .ªÄ4A,AXŽ.²:Ž, dGa࠻b .s␲ Ždmab ., where X,YgC,dgAYŽ.,and ␲ Žd m ab .gAX ŽmY .. Proof. Follows from MoritaŽ. Theorem 2.1 andwx 23, Theorem 5.1 . An algebra A in C satisfying conditionŽ. 1 or Ž. 2 is called a C-Azumaya algebra Ž.or an Azumaya algebra in C .

COROLLARY 3.2. Let A be an Azumaya algebra in C. Then the functors

A m ᎐ and ym A are monoidal functors and AAe C, Ce are monoidal categories.

Proof. Let X be an object in AAe C. Then X is an object in CAvia the composition maps

; Am X ª Ž.A࠻I m X ¨ ŽA࠻A .m X ª X,7Ž.

1 ␾y ; XmAªAmXªŽ.I࠻AmX¨ ŽA࠻A .mXªX.8Ž.

On the other hand, each object Y gAAC may be viewed as an object in AeCvia

idm␾ 6 Ž.A࠻A m YAmYmAªY.9Ž.

Thus one has a tensor product mAAin e C and A is a unit with respect to mAAwhich makes e C into a monoidal category. Now A m ᎐ is a monoidal functor between C and Ae C because, for M, N g C, the isomorphism

Ž.Ž.A m M mAAA m N , A m Ž.M m N g e C BRAIDED MONOIDAL CATEGORY 107 is as follows:

Similarly, ym A is a monoidal functor and Ce A is a monoidal category.

THEOREM 3.3. Let C be a braided monoidal category.

Ž.1wxP,P is an Azumaya algebra in C if P is faithfully projecti¨einC. Ž.2 If A is an Azumaya algebra in C, so is A. Ž.3 A࠻B is an Azumaya algebra in C if A, B are Azumaya algebras in C.

Proof. Ž.1 Let A be wxP, P . It is obvious that A is faithfully projec- UU UU tive. By Theorem 2.1 we have wxP, P ( Ä4P , P and A ( wxP , P . This UU UU yields that A࠻A ( wxwxwP, P ࠻ P , P ( P m P , P m P xwxs A, A , and UU UU A࠻A(Ä4Ä4ÄP,P࠻P,P(PmP,PmP4Ä4sA,A. Therefore A is an Azumaya algebra in C. Ž.2LetAbe an Azumaya algebra in C.Then A࠻A( A࠻A(Ä4A,A(wxwxA,A(A,A, and A࠻A ( A࠻A( wxA, A ( Ä4A, A (Ä4A,A. This means that A is Azumaya in C. Ž.3 Suppose that A, B are Azumaya in C. We have

Ž.A࠻B ࠻A࠻B , A࠻B࠻B࠻A ,A࠻wxB,B࠻A ,A࠻A࠻wxB,B ,wxwxA,A࠻B,B ,wxA࠻B,A࠻B.

Similarly, A࠻B࠻Ž.A࠻B ( Ä4A࠻B, A࠻B . 108 VAN OYSTAEYEN AND ZHANG

Now we are able to define the Brauer group of a braided monoidal category C. Let BŽ.C be the set of isomorphism classes of Azumaya algebras in C. Then BŽ.C is a semigroup with product ࠻ as before. Define a relation ; in BŽ.C as follows: A ; B if and only if there exist faithfully projective objects M, N g C such that

A࠻wxM,M(B࠻ wxN,N.1Ž.0

It is clear that ; is an equivalence relation in BŽ.C . Now we have

THEOREM 3.4. If C is a braided monoidal category, then the quotient set BŽ.Cr;is a group with product induced by ࠻ and an in¨erse operator induced by the opposite . This group is denoted by BrŽ.C and is called the Brauer group of category C. If A is C-Azumaya, wxA indicates the Brauer class in BrŽ.C . Remark 3.5.Ž. 1 So far we do not assume the braided monoidal categories considered to be additive. The algebras in the categories may be nonadditive. Thus theŽ. abelian Brauer group of a symmetric category defined inwx 22 is the special case of the Brauer groupŽ not necessarily abelian. of a braided category. Ž.2 The equivalence relationŽ. 10 turns out to be the Morita equivalence relation. One may generalize the proofs ofwx 5, 3.8᎐3.10 to obtain that C-Azumaya algebras A and B are equivalent if and only if A and B are Morita equivalent. Therefore we have that wxA s 1 g BrŽ.C if and only if A s wxP, P for some faithfully projective object P g C.

EXAMPLE 3.6. Let C be the symmetric monoidal category M k with the usual tensor product mk where k is a field or a commutative ring. It is easy to see that BrŽ.C s Br Ž. k , the classical Brauer group of k; cf.wx 2 .

EXAMPLE 3.7. Let Ž.X, OXXbe a scheme with ring structure sheaf O . Let C be the category of OX-module sheaves. Then C is a symmetric category with the usual tensor product over OX . It is a routine check that a sheaf of modules M in C is faithfully projective if and only if M is of finite type and locally free. Therefore a C-Azumaya algebra is exactly a sheaf of Azumaya algebras over OXX. So we have BrŽ.C s Br Ž O .; cf.w 1, 20x .

EXAMPLE 3.8. Let R be a commutative ring graded by an abelian group G. Let C be the category of graded R-modules. Then C is a symmetric category withŽ. graded tensor product over R. An algebra in C is a G-graded R-algebra. A C-Azumaya algebra is nothing else but a G-graded

R-Azumaya algebra. It follows that BrŽ.C s BrG Ž. R ; cf.wx 3 . BRAIDED MONOIDAL CATEGORY 109

EXAMPLE 3.9. Let C be the graded category Gr᎐R graded by the group ޚ2 , where R is a commutative ring with trivial gradation and the tensor product is over R. Define a braiding in C:

␾ : M m N ª N m M, where ␾Ž.m11m n syn 1mm 1if m 1g M 111, n g NMŽsM 0[M 1,N sN01[N., otherwise, ␾ s tw, the twist map. C together with ␾ is a braided monoidal category. The Brauer group BrŽ.C is nothing else but the Brauer᎐Wall group BWŽ. R ; cf.wx 29 .

EXAMPLE 3.10. Let G be an abelian group, R a commutative ring with trivial gradation. Let ␹ : G = G ª R be a bicharacter. Take C as the category of G-graded R-modules. Then Ž.C, mR , R, ␾ is a braided monoidal category, where ␾ is a braiding

M m N ª N m M, mghm n ¬ n hm m g␹ Ž.h m g , g, h g G.

One may easily see that the Brauer group of C is the Brauer group Br␹ Ž R, G. defined by Childs, Garfinkel, and Orzech inwx 6 . When ␹ is trivial, C is a symmetric category and BrŽ.C is the Brauer group Br Ž R, G .defined by Knus; cf.wx 10 .

EXAMPLE 3.11. Let H be a commutative and cocommutative Hopf algebra over a commutative ring k.An H-dimodule M is an R-module with a left H-module structure and a right H-comodule structure which satisfy the compatibility condition:

␳ Ž.h и m s Ýh и mŽ0.Žm m 1., where the notation Ý mŽ0.Žm m 1.denotes the comodule structure of ␳Ž.m , mgM. Let C be the category of H-dimodules and morphisms. Define a braiding in C as follows:

␾ : M m N ª N m M, m m n ¬ Ý mŽ1.Žи n m m 0..

Then Ž.C, mk , k, ␾ is a braided monoidal category and the Brauer group of Cis the Brauer᎐Long group BDŽ. H, k ; cf.wx 12, 13 . There are two braided monoidal subcategories of C. One is the subcategory of left H-modules

Ž.with trivial comodule structures H M, and another is the subcategory of right H-comodulesŽ. with the trivial left H-actions M H. In both subcategories the braiding ␾ becomes the twist map. So the Brauer groups H BrŽ.H M and Br ŽM .Žare abelian subgroups of Br C.. The most interesting Brauer group is the Brauer group of a Hopf algebra with bijective antipode. For the details, we refer towx 4, 5 . 110 VAN OYSTAEYEN AND ZHANG

EXAMPLE 3.12. Let H be a Hopf algebra over a commutative ring k with a bijective antipode S.An H-crossed module is a k-module M on which H acts on the left side and coacts on the right side. The action and coaction satisfy the compatibility condition: m g M, h g H,

y1 ␳ Ž.h и m s ÝhŽ2.Žи m 0.Žm hmS3.Ž1.ŽŽ.h1..

Let C be the category of H-crossed modules and their morphisms. Then

Ž.C,mk ,k,␾is a braided monoidal category where ␾ is defined as

M m N ª N m M, m m n ¬ Ý nŽ0.Žm n 1.и m, m g M, n g N.

The Brauer group of C is called the Brauer group of H-crossed module algebras, denoted by BQŽ. H, k . This notion of Brauer group contains the Brauer᎐Long group as a special case. However, it is quite different from the classical Brauer groups, even the Brauer᎐Long group, and it essen- tially deals with the finite representations and structure of the Hopf algebra. For instance, let H be a finite-dimensional Hopf algebra over a field and let AutŽ.H be the Hopf algebra automorphism group of H.We have an exact sequence of groupsŽ cf.wx 28. :

U 1 ª GDHŽ.Ž.Ž.ªGDH Ž.ªAut Ž.H ª BQ Ž k, H ., where GŽ.᎐ is the group functor from Hopf algebras to the groups of grouplike elements of the Hopf algebras and DHŽ.is the quantum double of H; cf.wx 7 . For example, if H is the Radford Hopf algebra of dimension nq1 m2,m)2, then GLnmŽ.ރ rU is a subgroup of BQŽރ, H .Žcf.wx 28. , where Umnis the subgroup of GL Ž.ރ which is isomorphic to the group of mth roots of units by diagonally embedding. This example shows that the Brauer group BQŽ. k, H may be a nontorsion group even if the Hopf algebra is ‘‘small.’’ Now let H be a quasitriangular Hopf algebra. That is, H is a Hopf Ž1. Ž2. algebra with an element R s Ý R m R in H m H satisfying several axiomsŽ cf.wx 18. . It is well known that the category H M of left H-modules is a braided monoidal category with a braiding ␾ as follows:

2 1 M m N ª N m M, m m n ¬ Ý R и n m R и m, m g M, n g N.

The Brauer group of H M is denoted by BMŽ. H, k which is a subgroup of BQŽ. H, k and contains the usual Brauer group BrŽ. k ; cf.wx 5 . On the other hand, if H is a coquasitriangularŽ. CQT Hopf algebra U ŽŽ.which is a pair H, R , here H is a Hopf algebra, R g Ž.H m H is invertible with respect to the convolution product and satisfies several axioms; cf.wx 11, 17. , the category M Hof right H-comodules is a braided BRAIDED MONOIDAL CATEGORY 111 monoidal category with braiding

M m N ª N m M,

m m n ¬ ÝnŽ0.Žm m 0.ŽRŽ.n 1.Žm m 1., m g M, n g N.

The Brauer group BrŽM H.Žis denoted by BC H, k.which also contains the usual Brauer group BrŽ. k and is a subgroup of BQŽ. H, k ; cf.wx 5 . Examples 3.9 and 3.10 are special cases of BCŽ. k, H when H is the group

Hopf algebra Rޚ2 and kG, respectively. Remark 3.13. Recently Drinfel’d and Majid generalized Hopf algebras to quasi-Hopf algebras and coquasi-Hopf algebras by deforming the co- multiplication or multiplication of Hopf algebras, and then they defined the quasitriangular quasi-Hopf algebra and coquasitriangular coquasi-Hopf algebraswx 8, 15 . For instance, if H is a quasitriangular quasi-Hopf algebra, then the category H M of left H-modules is a braided monoidal category. Therefore the Brauer group of H M may be constructed in the preceding way. To end this section, we present a generalized Rosenberg᎐Zelinsky sequence for an Azumaya algebra in a braided monoidal category C. Let us first define the Picard group of C. An object P in C is called an in¨ertible object if P m ᎐ defines an equivalence between C and itself. That is, there exists an object Q g C such that P m Q , I. The set of isomorphism classes of invertible objects in C becomes a group with the product m and unit represented by I. The Picard group of C is denoted by PicŽ.C . From the definition, one may easily see that Pic Ž.C contains the usual Picard group PicŽ.I . In all of the preceding examples Pic ŽC .is a direct product of PicŽ.I with another group; cf.wx 5, Prop. 4.3 . Now let A be an algebra in C. A morphism f: A ª A in C is said to be aC-automorphism if f is an algebra isomorphism in C. Denote by AutŽ.A the C-algebra automorphism group of A. An element ␣ g AutŽ.A is said 1 to be inner if there exists an element a g AIŽ.such that ␣ Ž.x s axay for any x g AXŽ.,XgC. Note that the set AI Ž.is a monoid with unit being the unit map ␮.Soay1 makes sense if it exists. It is obvious that the set of all inner elements of AutŽ.A is a subgroup of Aut Ž.A , denoted by

INNŽ.A . Given two C-automorphisms ␣, ␤ of A, we define an object ␣ A␤ e in Ae C as follows: ␣␤A s A as an object in C, but with A -module structure given by

␣m␤mid 6 idm␲ 6 ␲ Ž.A࠻A m AAmAmAAmAªA. 112 VAN OYSTAEYEN AND ZHANG

Now a straightforward verification shows that such objects as defined previously satisfy the following relations: for ␣, ␤, ␥ g AutŽ.A ,

Ž.i ␣ A␤␥␣␥␤, A Žvia ␥ .,

Ž.ii ␣ A␤ mA ␣␥A , ␣␤␥A ,

Ž.iii id A␣ ,idA id if and only if ␣ g INNŽ.A .

Given an ␣ g AutŽ.A , we have a canonical isomorphism in AeC:

id A␣␣, A m I ,

A where I␣ is an object in C. Let Ž.᎐ be the inverseŽ. equivalence functor A of A m ᎐ in Theorem 3.1. It is clear that Ž.id A␣␣s I . Since A m ᎐ is a monoidal functor, Ž.᎐ Ais a monoidal functor, too. It follows from relation Ž.ii that, in C,

AAA I␣␤sŽ.id A␣␤ ,Ž.id A␣ m Ž.id A␤␣␤,ImI,␣,␤gAutŽ.A .

It is easy to see that I␣ , ␣ g AutŽ.A , is an invertible object in C with inverse object I␣y1 . This yields a well-defined homomorphism ⌽ from the group AutŽ.A to the Picard group PicŽ.C , which is given by ⌽Ž.␣ s I␣ . Finally, relationŽ. iii gives rise to the following:

PROPOSITION 3.14. Let A be an Azumaya algebra in C. The following sequence is exact:

⌽ 1 ª INNŽ.A ª Aut Ž.A ª Pic Ž.C .11Ž.

4. SEPARABLE ALGEBRAS IN A BRAIDED MONOIDAL CATEGORY

In this section we study separable algebras in a braided monoidal category and construct a second Brauer group in the sense of separable algebras. In order to simplify our computations, we assume the braided monoidal category C to be a k-linear cocomplete abelian category with the unit k being a commutati¨e ring and with the ordinary tensor product o¨er k. Algebras in C are k-algebras with some extra structures; objects in C are k-modules with additional structures. Thus, in this section, we may use 1 to substitute the unit map ␮: k ª A for an algebra A in C and replace AkŽ. by A. BRAIDED MONOIDAL CATEGORY 113

An algebra A in C is said to be separable if the multiplication map

e ␲ A s A࠻A ª A

splits in Ae C. PROPOSITION 4.1. Let A be an algebra in C. The following are equi¨alent: Ž.1 A is separable. e Ž.2 There is an element clllg AŽ.Ž.Ž. k such that ␲ c s 1 and a࠻1 c s e Ž.1࠻aclgAk Ž.. Ž.3 There is an element c g A࠻AŽ.Ž.Ž. k such that ␲ c s 1 and a࠻1 c scŽ.1࠻agA࠻Ak Ž.. e Ž.4 There is an element crrg AŽ.Ž.Ž. k such that ␲ c s 1 and cr a࠻1 s e crŽ.1࠻agAk Ž.. ␲ Ž.5 The epimorphism A࠻A ª A splits in AAC . ␲ Ž.6 The epimorphism A࠻A ª A splits in CeA. y1 e Proof. One may choose cl as the inverse image of ␲ Ž.1 g A and c s crls c . Then a straightforward argument finishes the proof. The element c g A࠻AkŽ.satisfying the previous conditions is called a Casimir element of A. Each Casimir element c of A induces a map

Tr: C Ž.M, N ªAC Ž.M, N or Tr: C Ž.Ž.M, N ª C M, N A

for any two objects M, N g CA. Precisely, let X g C, c s a࠻b g A࠻AkŽ.. The trace map induced by c is

F ¬ Ä4MXŽ.2m¬fmab Ž .gNX Ž.gC ŽM,N .A.

Since the trace map is natural in X, we have

Tr: C Ž.Ž.X m M, N ª C X m M, N A.

This induces the trace map

Tr: wxwxM, N ª M, N A ,

where wxwxM, N , M, N A always exist because of our assumption on C. Similarly, for M, N gAC, we have a trace map

Tr: Ä4Ä4M, N ªA M, N . 114 VAN OYSTAEYEN AND ZHANG

Now since ␲ Ž.c s ab s 1, we even obtain that

Tr AAC Ž.Ž.M, N ª C M, N ª C Ž.M, N

is the identity on ACŽ.M, N . Similarly, the composition

Tr AAwxwxwxM,NªM,NªM,N

is the identity on AwxM, N . Now we consider the opposite and the product of separable algebras in C. To simplify the notation, we will write in the sequel, for example, u࠻¨ Ž.instead of Ýuii࠻¨ for a Casimir element cAof a separable algebra A.

PROPOSITION 4.2. Let A, B be separable algebras in C. Then

Ž.1 A is a separable algebra in C. Ž.2 A࠻B is a separable algebra in C.

Proof. Ž.1 Let cA s u࠻¨ be a Casimir element of A. Then a routine 1 computation shows that ␾y Ž.u m ¨ g A࠻AI Ž.is a Casimir element of A. So A is separable in C.

Ž.2 Let cABs u࠻¨ and c s x࠻y be the Casimir elements of A and B, respectively. Write c for the element ␾Ž.Ž.Ž.x m u ࠻ ¨࠻y g A࠻B ࠻Ž.Ž.A࠻Bk. We verify that c is a Casimir element of A࠻B. Write a࠻b for an arbitrary element of A࠻BXŽ.. We show that wŽ.Ž.a࠻b ࠻ 1࠻1 xc s cwŽ.Ž.1࠻1࠻a࠻bxwdiagrammatically. Ž.Ž.a࠻b ࠻ 1࠻1 xc s BRAIDED MONOIDAL CATEGORY 115 swŽ.Ž.a࠻1࠻1࠻1xwcŽ.Ž.1࠻1࠻1࠻bx. On the other hand, we have cwŽ.Ž.1࠻1࠻a࠻bxs

swŽ.Ž.a࠻1࠻1࠻1xwcŽ.Ž.1࠻1࠻1࠻bx. The proof is completed.

Note that in the precedingŽ. and later diagrams, the labeled lowercase elements are changed though we still use the same letters. We treat here the braiding of elements as we do the braiding of objects. One should understand that, for example, when element x passes a braiding it is no longer equal to x, but we still denote it by x.

DEFINITION 4.3. Let A be a separable algebra in C. The left center and l right center of A are defined respectively as follows: ZAŽ.sÄbgA<᭙a r gA,ab s ␲ Ž.a m b 4and ZAŽ.sÄbgA<᭙agA,ba s ␲ Ž.b m a 4.

PROPOSITION 4.4. Let A be a separable algebra in C. Then

Ž.1 ZAlŽ.,ZrŽ. A are objects in C. l r Ž.2AAeeÄ4A,AsZŽ. A andwx A, A s ZAŽ..

Proof. Ž.1 Let c be a Casimir element of A. Let ©, £ denote the left Ae-module and right eA-module structures of A, respectively. Then l r c©AsZAŽ.and A £ c s ZAŽ.. In fact, ᭙ x g A, let b s c © x. Then, for any a g A, ab s Ž.a࠻1 © b s Ž.a࠻1 c © x s Ž.1࠻ac©xs l Ž.1࠻a©bs␲ Žamb .. This implies that c © A : ZAŽ.. l Conversely, write c s a࠻b, ᭙ x g ZAŽ.,c©xsa␲ Žbmx .sabx s x. l r ZAŽ.:c©Afollows. The proof ZAŽ.sA£cis similar. 116 VAN OYSTAEYEN AND ZHANG

Second, we show that c © A and A £ c are objects in C. Let ␰ be the composition map of the unit map followed by the splitting map of ␲ :

k ª A ª A࠻A. e Denote by Ie the image of ␰ in A s A࠻A which is isomorphic to k. The e monomorphism Ie ª A induces a composition map in C:

e © Ie m A ª A m A ª A.

It is easy to see that c © A is exactly the image of the preceding composition map, and hence is an object in C. Similarly, A £ c is an object in C, too. l r Ž.2 Now we show that AAeeÄ4A, A s ZAŽ.and wxA, A s ZAŽ..We establish the first identity; the second one follows in a similar way. By definition, we have to check that

l Ae C Ž.A m X, A , C Ž.X, Z Ž.A

l is natural in X g C. Define a map ⌿ from AeCŽ.ŽA m X, A to C X, ZAŽ.. as follows: ᭙f gAeCŽ.A m X, A , we define ⌿ Ž.f as the following composition map:

f X s k m X ª A m X ª A.

That ⌿ is well defined follows from c © ⌿Ž.Ž.fXs⌿ Ž.Žfc© ŽkmX .. s⌿Ž.ŽŽ.f␲cX .s⌿ Ž.Ž.fXfor any f gAeC ŽA m X, A .. The inverse of ⌿ is given by

y1 l ⌿ : C Ž.X, Z Ž.A ªAe C ŽA m X, A .,

id g 6 m l ␲ g ¬ ž/A m XAmZAŽ.ªA.

The naturality in X with respect to ⌿ is clear.

An algebra A in C is said to be left Ž.or right central if AeÄ4A, A Žor wxA,AeA.is equal to k. Let A be a separable algebra in C with a Casimir element c s a࠻b. Then A is leftŽ. or right central if and only if a␲ Ž.b m x gkfor all x g A ŽŽor ␲ x m ab .gkfor all x g A.. PROPOSITION 4.5. Let A, B be separable algebras in C.

Ž.1AAeeÄ4A,A,wxA,A are direct summands of A in C. Ž.2 If A is leftŽ. or right central, then A is rightŽ. or left central. Ž.3A࠻B is left Ž. or right central if A, B are leftŽ. or right central. BRAIDED MONOIDAL CATEGORY 117

Proof. Ž.1 By Proposition 4.4, the subobjects AAeeÄ4A, A , wxA, A of A exist. The trace map Tr: A (AAÄ4Ä4A, A ª eA, A is exactly the map induced by a Casimir element c s u࠻¨ g A࠻A as follows:

This trace map splits the monomorphism AeÄ4A, A ª A. Similarly, another trace map

AªwxA,AeA,xª␲Ž.xmu¨sx£c,xgA, splits the monomorphism wxA, A eA ª A. In particular, if A is leftŽ. or right central, then k is a direct summand of A.

Ž.2 Let cA s u࠻¨ be a Casimir element of A. By Ž. 1 of Proposition y1 4.5, the element cAs ␾ Ž.u m ¨ g A࠻A is a Casimir element of A.If A£cAAsc ©A, then A is right central. Indeed, for x g A, we have x£cAs

scAA©x, where the first identity holds because c g A࠻AkŽ., and the second one holds because the multiplication operator commutes with the braiding.

Ž.3 It is sufficient to compute the elements of the form cA࠻B © d, where d g A࠻B and cA࠻BAis the Casimir element stemming from c and cB , Casimir elements of A and B, respectively. Let cABs u࠻¨, c s x࠻y, and cA࠻Bs ␾Ž.Ž.x m u ࠻ ¨࠻y . Given any element d s a࠻b g A࠻B,we have

In order to show that this element is in k, we use diagram chasing. First, let us point out the following equality for Casimir elements:

Ž.u࠻1 ࠻¨࠻1 cA࠻BAs Ž.u࠻1 ࠻1࠻11 Ž.࠻1࠻¨࠻1c࠻B

sŽ.u࠻1࠻1࠻1 Ž.¨࠻1࠻1࠻1cA࠻B

sŽ.u¨࠻1࠻1࠻1cA࠻B

scA࠻B. This yields

An algebra is central if both AAeeÄ4A, A , wxA, A are equal to k. Before we establish the main result of this section, we introduce a few more notions. Let A be an algebra in C. A subobject ᑧ of A is called an ideal or C-ideal of A if ᑧ is an object in AAC in the usual way. That is, the multiplication morphisms restrict to A m ᑧ ª ᑧ and ᑧ m A ª ᑧ. In particular, any subobject of k, unit of C, is an ideal of k. Note that C is cocomplete. If ᑧ is an ideal of an algebra A in C, then there exists a maximal ideal M containing ᑧ.If ᑧis an ideal of algebra A, then Arᑧ is an algebra in C. An algebra is said to be simpleŽ. or C-simple if A has no ideal except A or 0.

PROPOSITION 4.6. Let A be a leftŽ. or right central separable algebra in C. A is simple if and only if the unit k is simple. Proof. Suppose that A is a simple algebra in C and ᑧ is an ideal of k. The image of ᑧ m A ª A, denoted by ᑧ A, is an ideal of A because ᑧ A s Aᑧ.SoᑧAsAby the assumption. Now let Tr be the trace map Aªkwith Tr(␮ s idŽŽ.. assured by 1 of Proposition 4.5 , where ␮: k ª A is the unit morphism. Then k s TrŽ.A s Tr Žᑧ A .s ᑧ Tr Ž.A s ᑧ k s ᑧ. It follows that k is simple. Conversely, if k is simple, then k is a field because k is commutative.

Note that A is a separable algebra over k since cA, the Casimir element, is in A࠻AkŽ..So Ais a semisimple k-algebra. Now if ᑛ is a C-ideal of A, then ᑛ is a real ideal of A, and hence there is a central idempotent e g A such that ᑛ s Ae. We show that e is in the leftŽ. right center of A. This 120 VAN OYSTAEYEN AND ZHANG follows from the diagrams:

that is, for any a g A, ea s ␲ Ž.a m e . But ea s ae, and hence e is in l ZAŽ.sk. It follows that ᑛ l k / 0, which is an ideal of k, and hence ᑛ = k. It follows that A s ᑛ and A is simple. LEMMA 4.7. Let A be a leftŽ. or right central separable algebra in C, and let B be an algebra in C. Ž.1 If f: A ª B is an algebra epimorphism in C, then B is a separable algebra in C. Ž.2 AmK is a left Ž or right . central separable algebra in C m K s Ä X m KX< gC4 if K is a commutati¨ek-algebra. Proof. Ž.1 One may check that the algebra morphism f m f maps the Casimir elements of A to the Casimir elements of B.So Bis a separable algebra in C. Ž.2Ifcis a Casimir element of A, then c m 1 is a Casimir element l of A m K. The left center ZAŽ.Ž.Ž.mKscm1©AmKskm KsK. LEMMA 4.8. If A is a leftŽ. or right central separable algebra in C, then for any maximal C-ideal ᑧ of A, there exists a maximal ideal ᑛ of k such that ᑧ s ᑛ A and ᑧ l k s ᑛ. X Proof. Let ᑛ s ᑧ l A. Denote by C the category Ä4XgC<ᑛmXs0, X g C . X C is a braided monoidal category with the tensor product over krᑛ and X unit krᑛ. It is clear that A s Arᑧ is a simple algebra in C . Since Arᑧ is separable in C, it is separable in CX. Let c be a Casimir element of A. Then c, the image of c in Arᑧ࠻Arᑧ, is a Casimir element of Arᑧ. l The left center ZAŽ.rᑧsc©Arᑧs Žkqᑧ .rᑧskrᑛis the unit X X of C . Arᑧ is left central in C . Now, by Proposition 4.6, krᑛ is simple. It follows that ᑛ is a maximal ideal of k. On the other hand, Arᑛ A is a X separable algebra in C and the left center is c © Arᑛ A s krk l ᑛ A BRAIDED MONOIDAL CATEGORY 121 s krᑛ. Again, by Proposition 4.6 and the previous argument, Arᑛ A is simple. ᑛ A is a maximal ideal contained in ᑧ, and hence ᑧ s ᑛ A. Now we are able to show the main theorem of this section.

THEOREM 4.9. Let A be an algebra in C. The following are equi¨alent: Ž.1 A is a central separable algebra in C. Ž.2 A is a progenerator in C, and the following canonical morphisms are isomorphisms: Ž.iF:A࠻AªwxA,A,FaŽ.²:Ž࠻bcsa␲bmc ., Ž.ii G: A࠻A ª Ä4A, A , ²:cGa Ž࠻b .s␲ Žcmab .. Ž.3 A is separable and Azumaya in C.

Proof. Ž.1 m Ž.3 Suppose that A is central separable in C. By Proposi- tion 4.4, AewxA, A s k, and there exists a canonical Morita context in C:

e e X Ä4A , k, A, Ae w A, A x s A , f , g ,

Xe X where f: A m A ª A , a m p ª ²:apand g: A mAAeeA ª k s wxA, A , pmaªŽŽ²:..xªxp©a. e Let ␺ be the inverse map ␲ : A ª A such that ␲␺ s idA . Then X ␺gAkŽ.. For any x g A, x s ␲␺Ž.x s ␺ Ž.x © 1. This means that Ä4␺ ,1 X gAmAkŽ.is a dual basis for AeA. It follows that g isŽ. rationally surjective. Now we show that f is surjective. Let T be the image of f in Ae. T is certainly a C-ideal of Ae. Write B for Ae, and let c be a Casimir element X of A. For a m p g A m A ,

So ²:apgBcB and T : BcB. On the other hand, let p be defined by ²:xps Žx࠻1 .c. Then c s ²:1 p g T and BcB : T. Thus T s BcB.If T/B, then there exists a maximal ideal ᑧ containing T such that ᑧ s ᑛ B, where ᑛ is a maximal ideal of k. Now applying the morphism ␲ , we have

A s ␲ Ž.BcB : ␲ Ž.Ž.ᑧ s ␲ ᑛ B s ᑛ A.

By Lemma 4.8 we have ᑛ s ᑛ A l k s A l k s k, and hence ᑧ s B. Contradiction! f then is surjective. By the Morita theoremŽ. Theorem 2.1 e we have A ( wxA, A canonically and A is faithfully projective. Similarly, ee if one starts with A, then one has A ( Ä4A, A canonically. Thus we have proved that A is an Azumaya algebra in C. 122 VAN OYSTAEYEN AND ZHANG

Conversely, suppose that A is Azumaya and separable in C. Since A is e faithfully projective and A ( wxA, A canonically, the canonical Morita XX XX context Ä Ae, k, A, wxA, k , f , g 4is strict; that is, f , g are rationally surjective. This yields that AewxA, A ( k. This means A is left central. Similarly, e A(Ä4A,Aand the faithful projectivity of A imply that A is right central. Therefore A is central separable in C. Ž.2m Ž.3 Suppose that Ž. 2 holds. Since a progenerator is faithfully projective, A is Azumaya in C by definition. We show that A has a U Casimir element. Let f11m a g A m AkŽ.such that fa11 ² :s1gk.De- U fine a g001g AkŽ.by gx ²:sfax ²1 :. Let a࠻b be the element corre- sponding to 1 m g0 under the isomorphism

e U A(wxA,A(AmA. Then we have

ab s a1b s 1 g011²:1 s 1 fa ² :s1.

Moreover, for x, y g A,

Ž.Ž.x࠻1 a࠻by²:sxa␲ Žb m y .s xg0²: y

sŽ.1࠻xgy²²::0 1

sŽ.Ž.1࠻xa࠻by²:.

This implies that Ž.Ž.Ž.Ž.x࠻1 a࠻b s 1࠻xa࠻bfor any x g A. It follows from Proposition 4.1 that a m b is a Casimir element of A, and hence A is separable. Finally, to showŽ. 3 « Ž.2 , it is enough to show that A is a progenerator in C. By definition, we have to find an element p m a g wxA, k m A such that pa²:s1gk. This immediately follows by taking a s 1 and p s ŽTr: AªAewxA,A(k.. Here we have to point out thatŽ. 2 m Ž.3 holds in general for any braided monoidal category.

LEMMA 4.10. If P is a progenerator in C, thenwx P, P is a central separable algebra in C. Proof. The proof is the same as the proof ofwx 22, Theorem 14 . By the preceding theorem it is sufficient to show that wxP, P is separable in C. Let p00mfbe the dual basis for P and f1m p 1be an element in P m wxP, k such that fp11²:s1gk. Identify A s wxP, P with P m wxP, k which has e the usual multiplication. Define a࠻b \ Ž.p0110m f ࠻p m f g Ak Ž.. Then it is just routine to verify that a࠻b is a Casimir element of A. BRAIDED MONOIDAL CATEGORY 123

Now we are able to construct the second Brauer group of the category of C. Note that the progenerators are faithfully projective. So all the properties in Section 2 are valid for progenerators. Let BXŽ.C be the set of isomorphism classes of central separable algebras in C. Define an equivalent ; in BXŽ.C as follows: A ; B if and only if there exists progenerators P, Q g C such that

A࠻wxP,P(B࠻ wxQ,Q.

Now we have the following:

X X THEOREM 4.11. The quotient set B Ž.C r; , denoted by Br Ž.C , is a group with multiplication induced by ࠻ and in¨erse operator . There is a X monomorphism from Br Ž.C into Br Ž.C . If k is a projecti¨e object in C, then X Br Ž.C s Br Ž.C . Proof. If k is projective in C, then an object is a progenerator if X and only if it is faithfully projective. Br Ž.C s Br Ž.C follows from Theorem 4.9.

Remark 4.12.Ž. 1 BrXŽ.C may be defined by the separable Azumaya algebras if the braided monoidal category C is arbitrary. This is because conditionsŽ. 2 and Ž. 3 in Theorem 4.9 are always equivalent and Proposi- tion 4.2 holds in general. In fact, except for Proposition 4.6, the whole theory in this section can be extended to the case of an arbitrary braided monoidal category. Ž.2 In case C is a braided k-linear abelian cocomplete category, the conditionŽ inwx 22, Theorem 12. of the existence of an element c m d m e gAmAmAkŽ.such that ac m dbe s 1 m 1 may be dropped.

EXAMPLES 4.13. In Examples 3.6 and 3.7, the two Brauer groups BrŽ.C and BrXŽ.C coincide. In Examples 3.8᎐3.10, if the order of group G Žor

ޚ2 . is invertible in R, then the two Brauer groups coincide. In Examples 3.11 and 3.12, if the Hopf algebras are semisimple and cosemisimple, then the two Brauer groups are equal; cf.wx 5, Theorem 3.25 .

5. BASE CHANGE

In this section we discuss the behavior of Brauer groups under base changes. This will allow us to link the Brauer group of a braided monoidal category with the Brauer group of a Hopf algebra. 124 VAN OYSTAEYEN AND ZHANG

LEMMA 5.1wx 22, Theorem 17 and Corollary 18 . Let F be a monoidal functor from monoidal categories C to D Ž.not necessarily braided . Suppose that P is finite in C andw FŽ. P , ᎐xexists in D.

Ž.1 F Ž P . is finite in D. Ž.2 The canonical morphism ⌽: FP Žwx,X.ªwFPŽ.,F Ž X .x is an isomorphism for all X g C, where ⌽ is induced by the composite morphism

; FŽ.ev FPwx,X mFPŽ.ªFPŽ. wx,XmPªFXŽ..

Ž.3 If P is a progenerator, then FŽ. P is a progenerator. Ž.4 If F preserës coequalizers, P is faithfully projectië, then FŽ. P is a faithfully projectië object in D.

THEOREM 5.2. If F is a tensor functor from braided monoidal categories Cto D such that, for all finite objects P g C, w FPŽ.,᎐xexists, then F XX induces a homomorphism F˜: Br Ž.C ª Br ŽD .. If, in addition, F preser¨es the coequalizers, then F˜ is a homomorphism from BrŽ.C to Br ŽD ..

Proof. Let A be an algebra in C. FAŽ.is clearly an algebra in D, with multiplication

␦ FŽ.␲ FAŽ.mFA Ž.ªFA ŽmA .ªFA Ž.

FŽ.␮ ; X and unit J ª FIŽ.ªFA Ž ..IfwxA gBrŽ. C , then FA Ž.is a progenerator X by Lemma 5.1. To show that w FAŽ.xgBr Ž.D , we have to verify that the canonical morphisms

FAŽ.࠻FA Ž.ªFA Ž.,FA Ž., FA Ž.࠻FA Ž.ªÄ4FA Ž.,FA Ž. are in fact isomorphisms. Since F commutes with the braiding ␾, we have FAŽ.sFAŽ.. Now it follows from the following commutative diagrams that the previous canonical morphisms are isomorphismsŽ note that A s A BRAIDED MONOIDAL CATEGORY 125 as objects.

FAŽ.FA Ž. 6 FAŽ.,FA Ž. m w 6 x

␦ ⌽ 6

FŽ.; 6 FAŽ.mAFA Žwx,A. 6 FAŽ.mFA Ž. Ä FAŽ.,FA Ž.4 6

␦ ⌽X 6

FŽ.; 6 FAŽ.mAFA ŽÄ4,A..

Finally, suppose that A ; B in C. There are two progenerators P, Q g C such that A࠻wxP,P(B࠻ wxQ,Q. Now we have the commutative diagram 6 FAŽ mwxP,PFB.ŽmwxQ,Q.

␦y1 ␦y1

6 6

;6 FAŽ.mFP Žwx,PF.ŽB.ŽmFQwx,Q.

idm⌽ idm⌽

6 6 6 FAŽ.mwFPŽ.,FP Ž.x FBŽ.mwFQŽ.,FQ Ž.x, which implies that FAŽ.;FB Ž.in D.SoFinduces a well-defined homomorphism

XX F˜:Br Ž.C ª Br ŽD ., wxA ª FAŽ.. Now if F preserves coequalizers, then F preserves faithfully projective objects. The same argument as before shows that the induced homomorphism F˜ can be extended to a homomorphism BrŽ.C ª Br ŽD .. The foregoing theorem states that BrŽ.᎐ is a group scheme from the category of braided monoidal categories to the category of groups which 126 VAN OYSTAEYEN AND ZHANG restrict to an abelian group scheme on the closed subcategory of symmetric categorieswx 22 . Let C be a braided monoidal category. We consider a subcategory of C which consists of all finite objects in C Žit is not necessarily an abelian category. , denoted by C ff. It is clear that C is a rigid and closedŽ. under tensor product braided monoidal subcategory. Since all the Azumaya algebras in C are finite objects and fall in C f,it follows that the Brauer group BrŽ.C equals Br ŽC f.. Let k be a commuta- f tive ring, and let M k be the rigid category of finitely generated projective k-modules, where k is a commutative ring.

THEOREM 5.3. Let C be a braided monoidalŽ. small category with a monoidal functor F: C ª M k Ž.k a commutatië ring which sends the finite f objects into M k and preserës coequalizers. Then there exists a coquasitriangu- H X lar Hopf algebra oër k and a homomorphism F˜: BrŽ.C ª Br ŽM .ŽŽ.Br C X H X ª Br ŽM .. which satisfy the uniërsal property: If H is a coquasitriangular f H X Hopf algebra with a tensor functor G: C ª M preserïng coequalizers, through which F factors followed by the forgetful functor, then there exists a HHX homomorphism ␮: BrŽ.M ª Br ŽM .such that G˜˜s ␮F. Ž.U Ž. Proof. Set H0 s [X g F FX mFX and H s H0r; , where F is the set of all finite objects and ; is an equivalence relation such that

U UU FŽ.Ž␰y .mx;ymFŽ.Ž.␰x

U U for all ␰ : X ª Y, x g FXŽ.,y gFYŽ..His a coquasitriangular Hopf algebra. We refer towx 17, 31 for the details. For each finite object X, FXŽ. is a right H-comodule with the comodule structure:

coevmid 6 U idmp␫ 6 FXŽ.sImFX Ž. FX Ž.mFX Ž.mFX Ž. FX Ž.mH,

U where ␫: FXŽ.mFXŽ.ªH00is the usual inclusion and p: H ª H is X f H the projection map. Thus the functor F induces a functor F : C ª M sending object X to the object FXŽ.which is now a right H-comodule. X The functor F is a tensor functorŽ seewx 17, Theorems 2.2᎐2.8. . Since X f X FXŽ.,XgC, is finitely generated projective as a k-module, wFXŽ.,᎐x exists in M H. Moreover, FX respects coequalizers because F does. It follows from the previous theorem that the tensor functor FX induces a fH homomorphism F˜: BrŽ.C ª Br ŽM .. X Following from the Tannaka᎐Krein theoremwx 17, 18 , the functor F has the universal property. If H X is a coquasitriangular Hopf algebra and there X is a tensor functor G: C f M H such that F factors through G followed ª X by the forgetful functor from M H to M k, then there exists a coquasitrian- BRAIDED MONOIDAL CATEGORY 127

X gular Hopf algebra map f: H ª H and the induced tensor functor 1 f HHX m X f˜:MªM,X¬XXž/ªXmHªXmH such that the following diagram is commutative:

Now the tensor functor f˜ preserves coequalizers, and hence it induces a HHX homomorphism ␮: BrŽ.M ª Br ŽM .such that G˜˜s ␮F. Finally, there exists a Brauer group theory of coalgebras in a braided category. For instance, one may choose the braided monoidal category C as a k-linear abelian cocomplete category, where k is a field. Of course, k need not be the unit of C. A basic example is BrŽM R ., where R is a cocommutative k-coalgebra; cf.wx 26, 27 . A significant modification of wx 26 yields a theory of Azumaya coalgebras in a braided monoidal category.

ACKNOWLEDGMENT

The authors are grateful to the referee for his valuable suggestions and comments.

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