Braidings on the category of bimodules, Azumaya algebras and epimorphisms of rings
aa r X i v : . [ m a t h . QA ] O c t BRAIDINGS ON THE CATEGORY OF BIMODULES, AZUMAYAALGEBRAS AND EPIMORPHISMS OF RINGS
A. L. AGORE, S. CAENEPEEL, AND G. MILITARU
Abstract.
Let A be an algebra over a commutative ring k . We prove that braidingson the category of A -bimodules are in bijective correspondence to canonical R-matrices,these are elements in A ⊗ A ⊗ A satisfying certain axioms. We show that all braidingsare symmetries. If A is commutative, then there exists a braiding on A M A if andonly if k → A is an epimorphism in the category of rings, and then the corresponding R -matrix is trivial. If the invariants functor G = ( − ) A : A M A → M k is separable,then A admits a canonical R-matrix; in particular, any Azumaya algebra admits acanonical R-matrix. Working over a field, we find a remarkable new characterizationof central simple algebras: these are precisely the finite dimensional algebras that admita canonical R-matrix. Canonical R-matrices give rise to a new class of examples ofsimultaneous solutions for the quantum Yang-Baxter equation and the braid equation. Introduction
Braided monoidal categories play a key role in several areas of mathematics like quantumgroups, noncommutative geometry, knot theory, quantum field theory and 3-manifolds.It is well-known that the category A M A of bimodules over an algebra A over a commu-tative ring k is monoidal. The aim of this paper is to give an answer to the followingnatural question: given an algebra A , describe all braidings on A M A . Besides the purelycategorical significance this problem is also relevant in noncommutative geometry wherebraidings on A M A are used to develop the theory of wedge products of differential formsor connections on bimodules. The question is not as obvious as it seems: a first attemptmight be to use the switch map to define the braiding, but this is not well-defined, evenin the case when A is a commutative algebra. However, there are non-trivial examplesof braidings on the category of bimodules. For example, let A = M n ( k ) be a matrixalgebra; then c M,N : M ⊗ A N → N ⊗ A M given by the formula c M,N ( m ⊗ A n ) = n X i,j,t =1 e ij n e ti ⊗ A m e jt Mathematics Subject Classification.
Key words and phrases. braided category, epimorphism of rings, Azumaya algebra, separable functor,quantum Yang-Baxter equation.A.L. Agore is Aspirant fellow of FWO-Vlaanderen. S.Caenepeel is supported by FWO projectG.0117.10 “Equivariant Brauer groups and Galois deformations”. A.L. Agore and G. Militaru are sup-ported by the CNCS - UEFISCDI grant no. 88/05.10.2011 “Hopf algebras and related topics”. is a braiding on the category of A -bimodules (see Example 2.9 for full detail). A firstgeneral result is Theorem 2.1, stating that braidings on the category of A -bimodules arein bijective correspondence with canonical R-matrices, these are invertible elements R in the threefold tensor product A ⊗ A ⊗ A , satisfying a list of axioms. In this situation,we will say that ( A, R ) is an algebra with a canonical R-matrix. Actually, this result isinspired by a classical result of Hopf algebras: braidings on the category of (left) mod-ules over a bialgebra H are in one-to-one correspondence with quasitriangular structureson H , these are elements R in the two-fold tensor product H ⊗ H satisfying certainproperties. We refer to [12, Theorem 10.4.2] for detail. The next step is to reduce thelist of axioms to two equations, a centralizing condition and a normalizing condition,and then we can prove in Theorem 2.2 that all braidings on a category of bimodules aresymmetries. In the situation where A is commutative, we have a complete classification: A admits a canonical R-matrix R if and only if k → A is an epimorphism in the categoryof rings, and then R is trivial, see Proposition 2.3.The invariants functor G = ( − ) A : A M A → M k has a left adjoint F = A ⊗ − . Weprove that G is a separable functor [13, 14] if and only if G is fully faithful and thisimplies that A admits a canonical R-matrix. The converse property also holds if A isfree as a k -module, and then the braiding on the category of A -bimodules is unique, cf.Theorem 2.6.Azumaya algebras were introduced in [1] under the name central separable algebras; amore restrictive class was considered earlier by Azumaya in [2]. Azumaya algebras arethe proper generalization of central simple algebras to commutative rings. The Brauergroup consists of the set of Morita equivalence classes of Azumaya algebras. There existsa large literature on Azumaya algebras and the Brauer group, see for example the refer-ence list in [4]. A is an Azumaya algebra if and only if G is an equivalence of categories,and then G is separable. Therefore the category of bimodules over an Azumaya algebrais braided monoidal, that is any Azumaya algebra admits a canonical R-matrix. R canbe described explicitly in the cases where A is a matrix algebra or a quaternion algebra,see Examples 2.9 and 2.10. Not every algebra with a canonical R-matrix is Azumaya;for example Q is not a Z -Azumaya algebra, but 1 ⊗ ⊗ Z → Q is an epimorphism of rings. Thus algebras with a canonical R-matrix can beviewed as generalizations of Azumaya algebras.Applying Theorem 2.6 to finite dimensional algebras over fields, we obtain a new char-acterization of central simple algebras, namely central simple algebras are the finite di-mensional algebras admitting a canonical R-matrix. As a final application, we constructa simultaneous solution of the quantum Yang-Baxter equation and the braid equationfrom any canonical R-matrix, see Theorem 2.13.1. Preliminary
Azumaya algebras.
Let k be a commutative ring and A a k -algebra. Unadorned ⊗ means ⊗ k . A M A is the k -linear category of A -bimodules. It is well-known that wehave a pair of adjoint functors ( F, G ) between the category of k -modules M k and thecategory of A -bimodules A M A . For a k -module N , F ( N ) = A ⊗ N , with A -bimodulestructure a ( b ⊗ n ) c = abc ⊗ n , for all a , b , c ∈ A and n ∈ N . For an A -bimodule M , RAIDINGS ON THE CATEGORY OF BIMODULES 3 G ( M ) = M A = { m ∈ M | am = ma, ∀ a ∈ A } ∼ = A Hom A ( A, M ). The unit η and thecounit ε of the adjoint pair ( F, G ) are given by the formulas η N : N → ( A ⊗ N ) A ; η N ( n ) = 1 ⊗ n ; ε M : A ⊗ M A → M ; ε M ( a ⊗ m ) = am = ma for all n ∈ N , a ∈ A and m ∈ M A . Recall that A is an Azumaya algebra if A is faithfullyprojective as a k -module, that is, A is finitely generated, projective and faithful, and thealgebra map(1) F : A e = A ⊗ A op → End k ( A ) , F ( a ⊗ b )( x ) = axb is an isomorphism. Azumaya algebras can be characterized in several ways; perhaps themost natural characterization is the following: A is an Azumaya algebra if and only if theadjoint pair ( F, G ) is a pair of inverse equivalences, see [11, Theorem III.5.1]. Anothercharacterization is that A is central and separable as a k -algebra. Separable functors.
Recall from [13] that a covariant functor F : C → D is calledseparable if the natural transformation F : Hom C ( • , • ) → Hom D ( F ( • ) , F ( • )) ; F C,C ′ ( f ) = F ( f )splits, that is, there is a natural transformation P : Hom D ( F ( • ) , F ( • )) → Hom C ( • , • )such that P ◦ F is the identity natural transformation. Rafael’s Theorem [14] statesthat the left adjoint F in an adjoint pair of functors ( F, G ) is separable if and only ifthe unit of the adjunction η : 1 C → GF splits; the right adjoint G is separable if andonly if the counit ε : F G → D cosplits, that is, there exists a natural transformation ζ : 1 D → F G such that ε ◦ ζ is the identity natural transformation. A detailed study ofseparable functors can be found in [5]. Braided monoidal categories.
A monoidal category C = ( C , ⊗ , I, a, l, r ) consists of acategory C , a functor ⊗ : C ×C → C , called the tensor product, an object I ∈ C called theunit object, and natural isomorphisms a : ⊗ ◦ ( ⊗ × C ) → ⊗ ◦ ( C × ⊗ ) (the associativityconstraint), l : ⊗ ◦ ( I × C ) → C (the left unit constraint) and r : ⊗ ◦ ( C × I ) → C (theright unit constraint). a , l and r have to satisfy certain coherence conditions, we referto [10, XI.2] for a detailed discussion. C is called strict if a , l and r are the identitieson C . McLane’s Coherence Theorem asserts that every monoidal category is monoidalequivalent to a strict one, see [10, XI.5]. The categories that we will consider are notstrict, but they can be treated as if they were strict.Let τ : C × C → C × C be the flip functor. A prebraiding on C is a natural transformation c : ⊗ → ⊗ ◦ τ satisfying the following equations, for all U, V, W ∈ C : c U,V ⊗ W = ( V ⊗ c U,W ) ◦ ( c U,V ⊗ W ) ; c U ⊗ V,W = ( c U,W ⊗ V ) ◦ ( U ⊗ c V,W ) .c is called a braiding if it is a natural isomorphism. c is called a symmetry if c − U,V = c V,U ,for all
U, V ∈ C . We refer to [10, XIII.1] for more detail.
A. L. AGORE, S. CAENEPEEL, AND G. MILITARU Braidings on the category of bimodules
Let A be an algebra over a commutative ring k and A M A = ( A M A , − ⊗ A − , A ) themonoidal category of A -bimodules. A ( n ) will be a shorter notation for the n -fold tensorproduct A ⊗ · · · ⊗ A , where ⊗ = ⊗ k . An element R ∈ A (3) will be denoted by R = R ⊗ R ⊗ R , where summation is implicitly understood. Our first aim is to investigatebraidings on A M A . Theorem 2.1.
Let A be a k -algebra. Then there is a bijective correspondence between theclass of all braidings c on A M A and the set of all invertible elements R = R ⊗ R ⊗ R ∈ A (3) satisfying the following conditions, for all a ∈ A : R ⊗ R ⊗ aR = R a ⊗ R ⊗ R (2) aR ⊗ R ⊗ R = R ⊗ R a ⊗ R (3) R ⊗ aR ⊗ R = R ⊗ R ⊗ R a (4) R ⊗ R ⊗ ⊗ R = r R ⊗ r ⊗ r R ⊗ R (5) R ⊗ ⊗ R ⊗ R = R ⊗ R r ⊗ r ⊗ R r (6) where r = r ⊗ r ⊗ r = R . Under the above correspondence the braiding c correspondingto R is given by the formula (7) c M,N : M ⊗ A N → N ⊗ A M, c
M,N ( m ⊗ A n ) = R nR ⊗ A mR for all M , N ∈ A M A , m ∈ M and n ∈ N .An invertible element R ∈ A (3) satisfying (2)-(6) is called a canonical R-matrix and ( A, R ) is called an algebra with a canonical R-matrix.Proof. A (2) is an A -bimodule via the usual actions a ( x ⊗ y ) b = ax ⊗ yb , for all a , b , x , y ∈ A . Let c : A M A × A M A → A M A × A M A be a braiding on A M A . For each M, N ∈ A M A , we have an A -bimodule isomorphism c M,N : M ⊗ A N → N ⊗ A M , thatis natural in M and N . Now consider c A (2) ,A (2) : A (3) ∼ = A (2) ⊗ A A (2) → A (3) ∼ = A (2) ⊗ A A (2) , and let R = c A (2) ,A (2) (1 ⊗ ⊗ c is completely determined by R . For M, N ∈ A M A , m ∈ M and n ∈ N , we consider the A -bimodule maps f m : A (2) → M and g n : A (2) → N given by the formulas f m ( a ⊗ b ) = amb and g n ( a ⊗ b ) = anb . From the naturality of c ,it follows that ( g n ⊗ A f m ) ◦ c A (2) ,A (2) = c M,N ◦ ( f m ⊗ A g n ) . (7) follows after we evaluate this formula at 1 ⊗ ⊗
1. Obviously c M,N ( ma ⊗ A n ) = c M,N ( m ⊗ A an ). Furthermore, c M,N ( am ⊗ A n ) = ac M,N ( a ⊗ A n ) and c M,N ( m ⊗ A na ) = c M,N ( a ⊗ A n ) a since c M,N is a bimodule map. If we write these three formulas down inthe case where M = N = A (2) , and m = n = 1 ⊗
1, then we obtain (2-4). c satisfies thetwo triangle equalities c M ⊗ A N,P = ( c M,P ⊗ A N ) ◦ ( M ⊗ A c N,P ); c M,N ⊗ A P = ( N ⊗ A c M,P ) ◦ ( c M,N ⊗ A P ) . RAIDINGS ON THE CATEGORY OF BIMODULES 5
The first equality is equivalent to R pR ⊗ A m ⊗ A nR = r R pR r ⊗ A mr ⊗ A nR , for all m ∈ M , n ∈ N and p ∈ P . If we take M = N = P = A (2) and m = n = p = 1 ⊗ R ⊗ R ⊗ ⊗ R = r R ⊗ R r ⊗ r ⊗ R . Applying (4), we find that(5) holds. In a similar way, the second triangle equality implies (6).We can apply the same arguments to the inverse braiding c − . We set S = S ⊗ S ⊗ S = c − A (2) ,A (2) (1 ⊗ ⊗ m ⊗ A n = ( c − N,M ◦ c M,N )( m ⊗ A n ) = c − N,M ( R nR ⊗ A mR ) = S mR S ⊗ A R nR S . Now take m = n = 1 ⊗ ∈ A (2) . Then we find1 ⊗ ⊗ S ⊗ R S R ⊗ R S ) = R S ⊗ R S ⊗ R S ) = R S ⊗ S ⊗ R S R ) = R S R ⊗ S ⊗ S R ) = S R ⊗ S R ⊗ S R . In a similar way, we have that R S ⊗ R S ⊗ R S = 1 ⊗ ⊗
1, and it follows that S ⊗ S ⊗ S is the inverse of R ⊗ R ⊗ R .Conversely, assume that R ∈ A (3) is invertible and satisfies (2-6). Then we define c using(7). Straightforward computations show that c is a braiding on A M A . (cid:3) Let c be a braiding on A M A and R the corresponding canonical R -matrix. Then c is asymmetry, if and only if S = R , this means that(8) R r ⊗ R r ⊗ R r = 1 ⊗ ⊗ R − = R ⊗ R ⊗ R . The next theorem shows that the list of equations satisfiedby an R -matrix from Theorem 2.1 can be reduced to two equations and furthermore, weprove that all braidings on the category of A -bimodules are symmetries. Theorem 2.2.
Let A be a k -algebra. Then there is a bijection between the set of canon-ical R -matrices and the set of all elements R ∈ A (3) satisfying (4) and the normalizingcondition (9) R R ⊗ R = R ⊗ R R = 1 ⊗ Furthermore, R is invariant under cyclic permutation of the tensor factors, (10) R = R ⊗ R ⊗ R = R ⊗ R ⊗ R , and we have the additional normalizing condition (11) R ⊗ R R = 1 ⊗ . In particular, every braiding on A M A is a symmetry.Proof. Let R be an R -matrix as in Theorem 2.1, i.e. R is invertible and satisfies (2-6). Multiplying the second and the third tensor factor in (6), we find that R = R ⊗ R r r ⊗ R r = R (1 ⊗ r r ⊗ r ). From the fact that R is invertible, it follows that1 ⊗ ⊗ ⊗ r r ⊗ r , and the first normalizing condition of (9) follows after wemultiply the first two tensor factors. On the other hand, if we apply the flip map on A. L. AGORE, S. CAENEPEEL, AND G. MILITARU the last two positions in (5) we obtain that R ⊗ R ⊗ R ⊗ r R ⊗ r ⊗ R ⊗ r R .Multiplying the last two positions we obtain: R = r R ⊗ r ⊗ R r R ) = r R R ⊗ r ⊗ r R = R ( R R ⊗ ⊗ R )As R is invertible it follows that R R ⊗ R = 1 ⊗
1, as needed.Conversely, assume now that R satisfies (4) and (9). We will show that R is a canonical R -matrix satisfying (8) and hence from the observations preceding Theorem 2.2 we obtainthat the braiding c corresponding to R is a symmetry. First we show that R is invariantunder cyclic permutation of the tensor factors. R ⊗ R ⊗ R ) = R r r ⊗ r R ⊗ R ) = R r ⊗ r R ⊗ r R ) = r ⊗ r R R ⊗ r R ) = r ⊗ r ⊗ r . This implies immediately that the central conditions (2-3) are also satisfied. Next weshow that (5-6) are satisfied. r R ⊗ r ⊗ r R ⊗ R ) = R ⊗ r ⊗ r R r ⊗ R ) = R ⊗ R r ⊗ r r ⊗ R ) = R ⊗ R ⊗ ⊗ R ; R ⊗ R r ⊗ r ⊗ R r ) = R ⊗ r R r ⊗ r ⊗ R ) = R ⊗ r r ⊗ R r ⊗ R ) = R ⊗ ⊗ R ⊗ R . Finally, we prove that (8) holds since R r ⊗ R r ⊗ R r ) = R r R ⊗ r ⊗ R r ) = R R ⊗ r ⊗ R r r ) = 1 ⊗ r ⊗ r r ) = 1 ⊗ r ⊗ r r ) = 1 ⊗ ⊗ . and the proof is finished. (cid:3) The commutative case is completely classified by the following result.
Proposition 2.3.
Let A be a k -algebra. Then: (1) If a monomial x ⊗ y ⊗ z is a canonical R -matrix, then it is equal to ⊗ ⊗ . (2) 1 ⊗ ⊗ is a canonical R -matrix if and only if u A : k → A is an epimorphismof rings. (3) If A is commutative, then ( A, R ) is an algebra with a canonical R-matrix if andonly if R = 1 ⊗ ⊗ and u A : k → A is an epimorphism of rings.Proof.
1. Let R = x ⊗ y ⊗ z be a canonical R -matrix. From (5-6), it follows that x ⊗ ⊗ y ⊗ z = x ⊗ yx ⊗ y ⊗ z and x ⊗ y ⊗ ⊗ z = x ⊗ y ⊗ zy ⊗ z. Since R is invertible, this implies that1 ⊗ ⊗ ⊗ ⊗ yx ⊗ ⊗ z and 1 ⊗ ⊗ ⊗ x ⊗ ⊗ zy ⊗ , and, multiplying tensor factors, we find that 1 ⊗ yx ⊗ z and 1 ⊗ x ⊗ zy . It thenfollows that yxz = xzy = 1, hence y is invertible with y − = xz . Finally x ⊗ y ⊗ z = x ⊗ y y − ⊗ z ( ) = x ⊗ yy − ⊗ zy = 1 ⊗ ⊗ .
2. If R = 1 ⊗ ⊗
1, then the three centralizing conditions (2-4) are equivalent to a ⊗ ⊗ a , for all a ∈ A , which is equivalent to u A : k → A being an epimorphism RAIDINGS ON THE CATEGORY OF BIMODULES 7 of rings, see [15].3. Assume that (
A, R ) is an algebra with a canonical R-matrix. Then: R ⊗ R ⊗ ⊗ R ) = r R ⊗ r ⊗ r R ⊗ R ) = r R ⊗ R r ⊗ r ⊗ R = X R r ⊗ R r ⊗ r ⊗ R . At the third step, we used the fact that A is commutative. From the fact that R isinvertible, it follows that R ⊗ R ⊗ ⊗ R = 1 ⊗ ⊗ ⊗ R = 1 ⊗ ⊗
1. The restof the proof follows from 2. (cid:3)
Remarks 2.4.
1. The notion of quasi-triangular bialgebroid was introduced in [7, Def.19]. Quasi-triangular structures on a bialgebroid are given by universal R-matrices,see [7, Prop. 20] and [3, Def. 3.15], and correspond bijectively to braidings on thecategory of modules over the bialgebroid [3, Theorem 3.16]. It is well-known that A e is an A -bialgebroid, with the Sweedler canonical coring as underlying coring, and A -bimodules are left A e -modules. Comparing our Theorem 2.1 with the (left handed)version of [3, Theorem 3.16] yields that canonical R-matrices for A correspond bijectivelyto universal R-matrices for the canonical bialgebroid A e . This leads to an alternativeproof of Theorem 2.1, if we identify the R-matrices from [7] with our R-matrices. This,however, is more complicated than the straightforward proof that we presented, that alsohas the advantage that it is self-contained and avoids all technicalities on bialgebroids.2. (5-6) can be rewritten as R = R R and R = R R in the algebra A (4) .3. It follows from Proposition 2.3 that there is only one braiding on the category of (left) k -modules, namely the one given by the usual switch map.Before we state our next main result Theorem 2.6, we need a technical Lemma. If M ∈ A M A , then A ⊗ M is a k ⊗ A -bimodule, and we can consider( A ⊗ M ) k ⊗ A = { X i a i ⊗ m i ∈ A ⊗ M | X i a i ⊗ am i = X i a i ⊗ m i a, for all a ∈ A } . If M = A (2) , then ( A ⊗ A (2) ) k ⊗ A is the set of elements R ∈ A (3) satisfying (4). We havea map α M : A ⊗ M A → ( A ⊗ M ) A , α M ( a ⊗ m ) = a ⊗ m . Lemma 2.5.
Let M be an A -bimodule. The map α M is injective if A is flat as a k -module, and bijective if A is free as a k -module.Proof. If A is flat, then A ⊗ M A → A ⊗ M is injective, and then α M is also injective.Assume that A is free as a k -module, and let { e j | j ∈ I } be a free basis of A . Assumethat x = P i a i ⊗ m i ∈ ( A ⊗ M ) k ⊗ A . For all i , we can write a i = P j ∈ I α ji e j , for some α ji ∈ k . Then x = P j ∈ I e j ⊗ (cid:0)P i α ji m i (cid:1) . Now x = X j ∈ I e j ⊗ (cid:0)X i α ji am i (cid:1) = X i a i ⊗ am i = X i a i ⊗ m i a = X j ∈ I e j ⊗ (cid:0)X i α ji m i a (cid:1) , hence P i α ji am i = P i α ji m i a , for all j ∈ I , and P i α ji m i ∈ M A . We conclude that x = P j ∈ I e j ⊗ (cid:0)P i α ji m i (cid:1) ∈ Im α M , and this shows that α M is surjective. (cid:3) A. L. AGORE, S. CAENEPEEL, AND G. MILITARU
In our next result we assume A to be flat over k . Hence the map α A (2) is injective andthis will allows us to identify the elements in A ⊗ ( A ⊗ A ) A with the elements in A ⊗ A ⊗ A satisfying (4). Theorem 2.6.
Let A be a flat k -algebra and consider the conditions: (1) ( F, G ) is a pair of inverse equivalences, that is, A is an Azumaya algebra; (2) The functor G = ( − ) A : A M A → M k is fully faithful; (3) the functor G = ( − ) A : A M A → M k is separable; (4) there exists R = R ⊗ R ⊗ R ∈ A ⊗ ( A ⊗ A ) A such that R R ⊗ R = 1 ⊗ ; (5) there exists a unique R = R ⊗ R ⊗ R ∈ A ⊗ ( A ⊗ A ) A such that R R ⊗ R = 1 ⊗ ; (6) there exists a braiding on A M A , that is, there exists R ∈ A (3) such that ( A, R ) is an algebra with a canonical R-matrix.Then (1) ⇒ (2) ⇔ (3) ⇔ (4) ⇔ (5) ⇒ (6) . If A is central, then (2) ⇒ (1) . If A is freeas a k -module, then (6) ⇒ (5) , and in this case the braiding on A M A is unique. If k isa field, and A is finite dimensional, then (6) ⇒ (1) , and all six assertions are equivalent.Proof. (1) ⇒ (2), (2) ⇒ (3) and (5) ⇒ (4) are trivial.(3) ⇒ (4). If G is separable, then we have a natural transformation ζ : 1 ⇒ F G suchthat ε M ◦ ζ M = M , for all M ∈ A M A . Now let R = ζ A (2) (1 ⊗
1) = R ⊗ R ⊗ R ∈ F G ( A (2) ) = A ⊗ ( A ⊗ A ) A . Then 1 ⊗ ε A (2) ◦ ζ A (2) )(1 ⊗
1) = R R ⊗ R .The natural transformation ζ is completely determined by R . For an A -bimodule M and m ∈ M , we define f m as in the proof of Theorem 2.1. From the naturality of ζ , itfollows that the diagram A (2) f m (cid:15) (cid:15) ζ A (2) / / A ⊗ ( A ⊗ A ) AA ⊗ ( f m ) A (cid:15) (cid:15) M ζ M / / A ⊗ M A commutes. Evaluating the diagram at 1 ⊗
1, we find that(12) ζ M ( m ) = R ⊗ R mR . (4) ⇒ (6). Write R = P i a i ⊗ b i , with a i ∈ A and b i ∈ ( A ⊗ A ) A . Then R ⊗ R R = P i b i a i = P i a i b i = R R ⊗ R = 1 ⊗
1, thus R satisfies (9). Moreover, as R ∈ A ⊗ ( A ⊗ A ) A it follows that R also satisfies (4). Then using Theorem 2.2 we obtain that R is a canonical R -matrix and it determines a braiding on A M A .(4) ⇒ (2). Given R ∈ A ⊗ ( A ⊗ A ) A satisfying R R ⊗ R = 1 ⊗
1, we define ζ using(12). It follows immediately that ( ε M ◦ ζ M )( m ) = ε ( R ⊗ R mR ) = R R mR = m .We have seen in the proof of 4) ⇒
6) that (3) and (11) are satisfied. For a i ∈ A and m i ∈ M A , we then compute( ζ M ◦ ε M )( X i a i ⊗ m i ) = X i R ⊗ R a i m i R ) = X i a i R ⊗ R m i R = X i a i R ⊗ R R m i ( ) = X i a i ⊗ m i . This shows that ε is a natural transformation with inverse ζ , and G is fully faithful.(2) ⇒ (5). We have already seen that 2) implies 4), and this shows that R exists. If G is fully faithful, then ε M is invertible, for all M ∈ A M A . If R ∈ A ⊗ ( A ⊗ A ) A satisfies R R ⊗ R = 1 ⊗
1, then ε A ⊗ A ( R ) = 1 ⊗
1, hence R = ε − A ⊗ A (1 ⊗ ⇒ (4). From (5), it follows that there exists R ∈ ( A ⊗ A (2) ) k ⊗ A such that R R ⊗ R =1 ⊗
1, see Theorem 2.2. α A (2) is bijective, see Lemma 2.5, hence α − A (2) ( R ) ∈ A ⊗ ( A ⊗ A ) A satisfies (3). The uniqueness of R follows from (4).(4) ⇒ (1). Assume that A is central. From (4), it follows that ε A ⊗ A : A ⊗ ( A ⊗ A ) A → A ⊗ A is surjective, and then it follows from [1, Theorem 3.1] that A is separable over Z ( A ) = k . Thus A is central separable, and therefore Azumaya.(6) ⇒ (1). If k is a field, then A is free, so (6) implies (5), and, a fortiori, (2). Then ε A : A ⊗ A A → A is an isomorphism of A -bimodules, and therefore also of vector spaces.A count of dimensions shows that dim k ( Z ( A )) = dim k ( A A ) = 1, so that Z ( A ) = k A ,and A is central, and then (1) follows from (2). (cid:3) In particular, applying Theorem 2.6 for finite dimensional algebras over fields we obtainthe following surprising characterization of central simple algebras:
Corollary 2.7.
For a finite dimensional algebra A over a field k , the following assertionsare equivalent: (1) A is a central simple algebra; (2) there exists a (unique) braiding on A M A ; (3) there exists a (unique) invertible element R ∈ A ⊗ A ⊗ A satisfying the conditions R ⊗ a R ⊗ R = R ⊗ R ⊗ R a and R R ⊗ R = R ⊗ R R = 1 ⊗ , for all a ∈ A . For any k -algebra A , the functor F : M k → A M A is strong monoidal. Indeed, for any N, N ′ ∈ M k , we have natural isomorphisms ϕ : F ( k ) = A ⊗ k → A and ϕ N,N ′ : F ( N ) ⊗ A F ( N ′ ) = ( A ⊗ N ) ⊗ A ( A ⊗ N ′ ) → F ( N ⊗ N ′ ) = A ⊗ N ⊗ N ′ satisfying all the necessary axioms, see [10]. Proposition 2.8.
Let ( A, R ) be an algebra with a canonical R-matrix. Then the functor F : M k → A M A preserves the symmetry.Proof. We have to show that the following diagram commutes( A ⊗ N ) ⊗ A ( A ⊗ N ′ ) c A ⊗ N,A ⊗ N ′ (cid:15) (cid:15) ϕ N,N ′ / / A ⊗ N ⊗ N ′ A ⊗ τ N,N ′ (cid:15) (cid:15) ( A ⊗ N ′ ) ⊗ A ( A ⊗ N ) ϕ N ′ ,N / / A ⊗ N ′ ⊗ N Here τ N,N ′ : N ⊗ N ′ → N ′ ⊗ N is the usual switch map. For a, b ∈ A , n ∈ N and n ′ ∈ N ′ , we compute( ϕ N ′ ,N ◦ c A ⊗ N,A ⊗ N ′ )(( a ⊗ n ) ⊗ A ( b ⊗ n ′ )) ( ) = ϕ N ′ ,N (cid:0) ( R bR ⊗ n ′ ) ⊗ A aR ⊗ n (cid:1) = R bR aR ⊗ n ′ ⊗ n ( ) = R R abR ⊗ n ′ ⊗ n ( ) = ab ⊗ n ′ ⊗ n = ab ⊗ τ N,N ′ ( n ⊗ n ′ )= (( A ⊗ τ N,N ′ ) ◦ ϕ N,N ′ )(( a ⊗ n ) ⊗ A ( b ⊗ n ′ ))and the proof is finished. (cid:3) If A is an Azumaya algebra, then it follows from Theorem 2.6 that we have a symmetry onthe category of A -bimodules A M A . In Examples 2.9 and 2.10, we give explicit formulasfor R in the case where A is a matrix ring or a quaternion algebra; in both cases A isfree, so that the canonical R -matrix is unique. Example 2.9.
Let A = M n ( k ) be a matrix algebra. Then the R -matrix for M n ( k ) ifgiven by R = n X i,j,k =1 e ij ⊗ e ki ⊗ e jk where e ij is the elementary matrix with 1 in the ( i, j )-position and 0 elsewhere. Indeed,for all indices i , j , p , q , we have e pq ( n X k =1 e ki ⊗ e jk ) = e pi ⊗ e jq = ( n X k =1 e ki ⊗ e jk ) e pq , hence P nk =1 e ki ⊗ e jk ∈ ( A ⊗ A ) A and R = P ni,j =1 e ij ⊗ ( P nk =1 e ki ⊗ e jk ) ∈ A ⊗ ( A ⊗ A ) A .Finally n X i,j,k =1 e ij e ki ⊗ e jk = X i,j =1 n e ii ⊗ e jj = 1 ⊗ Example 2.10.
Let K be a commutative ring, such that 2 is invertible in K , and taketwo invertible elements a, b ∈ K . The generalized quaternion algebra A = a K b is thefree K -module with basis { , i, j, k } and multiplication defined by i = a, j = b, ij = − ji = k. It is well-known that A is an Azumaya algebra. The corresponding R -matrix is R = 14 (1 ⊗ ⊗
1) + 14 a (1 ⊗ i ⊗ i + i ⊗ ⊗ i + i ⊗ i ⊗ b (1 ⊗ j ⊗ j + j ⊗ ⊗ j + j ⊗ j ⊗ − ab (1 ⊗ k ⊗ k + k ⊗ ⊗ k + k ⊗ k ⊗ ab ( i ⊗ j ⊗ k + j ⊗ k ⊗ i + k ⊗ i ⊗ j ) − ab ( j ⊗ i ⊗ k + k ⊗ j ⊗ i + i ⊗ k ⊗ j ) . It is easy to show that R satisfies (4) and (9). Indeed, R R ⊗ R = 14 (1 ⊗
1) + 14 a ( i ⊗ i + i ⊗ i + a ⊗ RAIDINGS ON THE CATEGORY OF BIMODULES 11 + 14 b ( j ⊗ j + j ⊗ j + b ⊗ − ab ( k ⊗ k + k ⊗ k − ab ⊗ ab ( k ⊗ k − bi ⊗ i − aj ⊗ j ) + 14 ab ( k ⊗ k − bi ⊗ i − aj ⊗ j ) = 1 ⊗ , proving the first normalization from (9); the second one follows in a similar manner. Anelementary computation shows that R ⊗ xR ⊗ R = R ⊗ R ⊗ R x , for x = i, j, k , andthis proves (4). Example 2.11.
Any Azumaya algebra A admits a canonical R-matrix. The converseis not true: it suffices to consider Q as a Z -algebra. Since Z ⊂ Q is an epimorphism ofrings, it follows from Proposition 2.3 that ( Q , ⊗ ⊗
1) is braided and it is obvious that Q is not a Z -Azumaya algebra. Example 2.12.
Let (
A, R ), (
B, S ) be two algebras with a canonical R-matrix. It isstraightforward to show that ( A ⊗ B, T ), with T := R ⊗ S ⊗ R ⊗ S ⊗ R ⊗ S ∈ ( A ⊗ B ) (3) ,is an algebra with a canonical R-matrix.We conclude this paper with another application of canonical R-matrices: they can beapplied to deform the switch map into a simultaneous solution of the quantum Yang-Baxter equation and the braid equation. Theorem 2.13.
Let ( A, R ) be an algebra with a canonical R-matrix and V an A -bimodule. Then the map Ω : V ⊗ V → V ⊗ V, Ω( v ⊗ w ) = R wR ⊗ R v is a solution of the quantum Yang-Baxter equation Ω Ω Ω = Ω Ω Ω and of thebraid equation Ω Ω Ω = Ω Ω Ω . Moreover Ω = Ω in End( V (2) ) .Proof. Let r = S = R . Then for any v , w , t ∈ V we have:Ω Ω Ω ( v ⊗ w ⊗ t ) = Ω Ω ( v ⊗ R t R ⊗ R w )= Ω ( r R w r ⊗ R t R ⊗ r v ) = S R t R S ⊗ S r R w r ⊗ r v ( ) = R t R S S ⊗ S r R w r ⊗ r v ( ) = R t R ⊗ r R w r ⊗ r v ( ) = R t R ⊗ r w r ⊗ R r v andΩ Ω Ω ( v ⊗ w ⊗ t ) = Ω Ω ( R w R ⊗ R v ⊗ t )= Ω ( r t r ⊗ R v ⊗ r R w R ) ) = r t r ⊗ S r R w R S ⊗ S R v ( ) = r t r ⊗ S R w R S ⊗ r S R v ( ) = r t r ⊗ R w R S S ⊗ r S R v ( ) = r t r ⊗ R w R ⊗ r R v, Hence Ω is a solution of the quantum Yang-Baxter equation. On the other hand:Ω Ω Ω ( v ⊗ w ⊗ t ) = Ω Ω ( R w R ⊗ R v ⊗ t )= Ω ( R w R ⊗ r t r ⊗ r R v ) = S r t r S ⊗ S R w R ⊗ r R v ( ) = r t r S S ⊗ S R w R ⊗ r R v ( ) = r t r ⊗ R w R ⊗ r R v and Ω Ω Ω ( v ⊗ w ⊗ t ) = Ω Ω ( v ⊗ r t r ⊗ r w )= Ω ( S r t r S ⊗ S v ⊗ r w ) = S r t r S ⊗ R r w R ⊗ R S v ( ) = r t r S S ⊗ R r w R ⊗ R S v ( ) = r t r ⊗ R r w R ⊗ R v ( ) = r t r ⊗ R w R ⊗ r R v. Thus Ω is also a solution of the braid equation. Finally,Ω ( v ⊗ w ) = S r R w R S ⊗ S r R v r ) = S w R S ⊗ r R S r R v r ) = S w S ⊗ r R S R r R v r ) = S w S ⊗ R S R r r R v r ) = S w S ⊗ R S R R v ( ) = S w S ⊗ S v = Ω( v ⊗ w ) . (cid:3) References [1] M. Auslander, O. Goldman, The Brauer group of a commutative ring,
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