The Brauer Group of a Hopf Algebra

New Directions in Hopf Algebras MSRI Publications Volume 43, 2002 The Brauer Group of a Hopf Algebra FREDDY VAN OYSTAEYEN AND YINHUO ZHANG Abstract. Let H be a Hopf algebra with a bijective antipode over a commutative ring k with unit. The Brauer group of H is defined as the Brauer group of Yetter–Drinfel’d H-module algebras, which generalizes the Brauer–Long group of a commutative and cocommutative Hopf algebra and those known Brauer groups of structured algebras. Introduction The Brauer group is something like a mathematical chameleon, it assumes the characteristics of its environment. For example, if you look at it from the point of view of representation theory you seem to be dealing with classes of noncommutative algebras appearing in the representation theory of finite groups, a purely group theoretical point of view presents it as the second Galois-cohomology group, over number fields it becomes an arithmetical tool related to the local theory via complete fields, over an algebraic function field or some coordinate rings it gets a distinctive geometric meaning and category theoretical aspects are put in evidence when relating the Brauer group to K-theory, in particular the K2 -group. When looking at the vast body of theory existing for the Brauer group one cannot escape to note the central role very often played by group actions and group gradings. This is most evident for example in the appearance of crossed products or generalizations of these. Another typical case is presented by Clifford algebras and the Z 2 (i.e., Z/2Z) graded theory contained in the study of the well-known Brauer–Wall group [57], as well as the generalized Clifford algebras in the Brauer–Long group for an abelian group [29]. At that point the theory was ripe for an approach via Hopf algebras where certain actions and co-actions (like the grading by a group) may be adequately combined in one unifying theory [30], but for commutative cocommutative Hopf algebras only. However, the cohomological interpretation for such This work was supported by UIA and the URC of USP. 437 438 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG Brauer–Long groups presented some technical problems that probably slowed down the development of a general theory. The cohomological description was obtained years later by S. Caenepeel a.o. [6; 7; 8; 9] prompted by new interest in the matter stemming from earlier work of Van Oystaeyen and Caenepeel, Van Oystaeyen on another type of graded Brauer group. The problem of considering noncommutative noncocommutative Hopf algebras remained and became more fascinating because of the growing interest in quantum groups. The present authors then defined and studied the Brauer group of a quantum group first in terms of the category of Yetter–Drinfel’d modules, but quickly generalized it to the Brauer group of a braided category [53], thus arriving at the final generality one would hope for after [38]. The Brauer group of a quantum group or even of a general Hopf algebra presents us with an interesting new invariant but a warning is in place. Not only is this group non-abelian, it is even non-torsion in general! Even restriction to cohomology describable or split parts does not reduce the complexity much. On the other hand, at least for finite dimensional Hopf algebras explicit calculations should be possible. Note that even the case of the Brauer group of the group ring of a non-abelian group is a very new and interesting object. Recently concrete calculations have been finalized for Sweedler’s four dimensional Hopf algebra, group rings of dihedral groups and a few more low dimensional examples [16; 54; 56]. The arrangement of this paper is as follows: (i) Basic notions and conventions (ii) Quaternion algebras (iii) The definition of the Brauer group (iv) An exact sequence for the Brauer group BC(k, H, R) (v) The Hopf automorphism group (vi) The second Brauer group We do not repeat here a survey of main results because the paper is itself an expository paper albeit somewhat enriched by new results at places. We have adopted a very constructive approach starting with a concrete treatment of actions and coactions on quaternion algebras (Section 2), so that the abstractness of the definition in Section 3 is well-motivated and is made look natural. We shall not include the Brauer–Long group theory in this paper as the reader may find a comprehensive introduction in the book [6]. 1. Basic Notions and Conventions Throughout k is a commutative ring with unit unless it is specified and (H, ∆, ε, S) or simply H is a Hopf algebra over k where (H, ∆, ε) is the underlying coalgebra and S is a bijective antipode. Since the antipode S is bijective, the opposite H op and the co-opposite H cop are again Hopf algebras with antipode THE BRAUER GROUP OF A HOPF ALGEBRA 439 S −1 . We will often use Sweedler’s sigma notation; for example, we will write, for h ∈ H, P i. ∆h = h(1) ⊗ h(2) , P ii. (1 ⊗ ∆)∆h = (∆ ⊗ 1)∆h = h(1) ⊗ h(2) ⊗ h(3) etc. For more detail concerning the theory of Hopf algebras we refer to [1; 35; 47]. 1.1. Dimodules and Yetter–Drinfel’d modules. A k-module is said to be of finite type if it is finitely generated projective. If a k-module M of finite type is faithful, then M is said to be faithfully projective Let H be a Hopf algebra. We will take from [47] the theory of H-modules and P H-comodules for granted. Sigma notations such as m(0) ⊗ m(1) for the comodule structure χ(m) of an element m of a left H-comodule M will be adapted from [47]. In this paper we will use χ for comodule structures over Hopf algebras and use ρ for comodule structures over coalgebras in order to distinguish two comodule structures when they happen to be together. Write H M for the category of left H-modules and H-module morphisms. If M and N are H-modules the diagonal H-module structure on M ⊗ N and the adjoint H-module structure on Hom(M, N ) are given by: P i. h · (m ⊗ n) = h(1) · m ⊗ h(2) · n, P ii. (h · f )(m) = h(1) · f (S(h(2) ) · m), for h ∈ H, n ∈ N, f : M −→ N . The category H M together with the tensor product and the trivial H-module k forms a monoidal category (see [31]). If M is left H-module, we have a k-module of invariants M H = {m ∈ M | h · m = ε(h)m.} In a dual way, we have a monoidal category of right H-comodules, denoted (MH , ⊗, k) or simply MH . For instance, if M and N are two right H-comodules, the codiagonal H-comodule structure on M ⊗ N is given by X χ(m ⊗ n) = m(0) ⊗ n(0) ⊗ m(1) n(1) for m ∈ M and n ∈ N . If an H-comodule M is of finite type, then Hom(M, N ) ∼ = N ⊗ M ∗ has a comodule structure: X χ(f )(m) = f (m(0) )(0) ⊗ f (m(0) )(1) S(m(1) ) for f ∈ Hom(M, N ) and m ∈ M . For a right H-comodule M the k-module M coH = {m ∈ M | χ(m) = m ⊗ 1} is called the coinvariant submodule of M . A k-module M which is both an H-module and an H-comodule is called an H-dimodule if the action and the coaction of H commute, that is, for all m ∈ M, h ∈ H, X X (h · m)(0) ⊗ (h · m)(1) = h · m(0) ⊗ m(1) . 440 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG Write DH for the category of H-dimodules. When the Hopf algebra H is finite, we obtain equivalences of categories ∗ MH⊗H ∼ =H ∗ ⊗H M; = DH ∼ here H ⊗H ∗ and H ∗ ⊗H are tensor Hopf algebras. For more details on dimodules, we refer to [29; 30]. Recall that a Yetter–Drinfel’d H-module (simply a YD H-module) M is a left crossed H-bimodule [58]. That is, M is a k-module which is a left H-module and a right H-comodule satisfying the following equivalent compatibility conditions [27, 5.1.1]: P P i. h(1) · m(0) ⊗ h(2) m(1) = (h(2) · m)(0) ⊗ (h(2) · m)(1) h(1) P ii. χ(h · m) = (h(2) · m(0) ) ⊗ h(3) m(1) S −1 (h(1) ). Denote by QH ( H Y DH in several references) the category of YD H-modules and YD H-module morphisms. For two YD H-modules M and N , the diagonal Hmodule structure and the codiagonal H op -comodule structure on tensor product M ⊗ N satisfy the compatibility conditions of a YD H-module. So M ⊗ N is a ˜ . It is easy to see that the natural map YD H-module, denoted M ⊗N ˜ )⊗Z ˜ −→ X ⊗(Y ˜ ⊗Z) ˜ Γ : (X ⊗Y is a YD H-module isomorphism, and the trivial YD H-module k is a unit with ˜ Therefore (QH , ⊗, ˜ k) forms a monoidal category (for details conrespect to ⊗. cerning monoidal categories we refer to [31; 58]). Let M and N be YD H-modules. Then there exists a YD H-module isomor˜ and N ⊗M ˜ : phism Ψ between M ⊗N X ˜ (1) · m ˜ −→ N ⊗M, ˜ ˜ 7→ Ψ : M ⊗N m⊗n n(0) ⊗n P ˜ ˜ (0) . It is not hard to check that with inverse Ψ−1 (n⊗m) = S(n(1) ) · m⊗n ˜ Γ, Ψ, k) is a braided monoidal (or quasitensor) category (see [31; 58]). If (QH , ⊗, in addition, H is a finite Hopf algebra, then there is a category equivalence: D(H) M ∼ QH where D(H) is the Drinfel’d double (H op )∗ ⊲⊳ H which is a finite quasitriangular Hopf algebra over k as described in [23; 32; 40]. 1.2. H-dimodule and YD H-module algebras. An algebra A is a (left) Hmodule algebra if there is a measuring action of H on A, i.e., for h ∈ H, a, b ∈ A, i. A is a left H-module, P ii. h · (ab) = (h(1) · a)(h(2) · b), iii. h · 1 = ε(h)1. Similarly, an algebra is called a (right) H-comodule algebra if A is a right Hcomodule with the comodule structure χ : A −→ A ⊗ H being an algebra map, i.e., for a, b ∈ A, THE BRAUER GROUP OF A HOPF ALGEBRA i. ii. 441 P χ(ab) = a(0) b(0) ⊗ a(1) b(1) , χ(1) = 1 ⊗ 1. An H-dimodule algebra A is an H-dimodule and a k-algebra which is both an H-module algebra and an H-comodule algebra. Suppose that H is both commutative and cocommutative. Let A and B be two H-dimodule algebras. The smash product A#B is defined as follows: A#B = A ⊗ B as a k-module and the multiplication is given by X (a#b)(c#d) = a(b(1) · c)#b(0) d. Then A#B furnished with the diagonal H-module structure and codiagonal comodule structure A ⊗ B is again an H-dimodule algebra. The H-opposite A of an H-dimodule algebra A is equal to A as an H-dimodule, but with multiplication given by X a·b= (a(1) · b)a(0) which is again an H-dimodule algebra. A Yetter–Drinfel’d H-module algebra A is a YD H-module and a k-algebra which is a left H-module algebra and a right H op -comodule algebra. Note that here we replace H by H op when we deal with comodule algebra structures. As examples pointed out in [11], (H op , ∆, ad′ ) and (H, χ, ad) are regular Yetter–Drinfel’d H-module algebras with H-structures defined as follows: X h ad′ x = h(2) xS −1 (h(1) ) X ∆(x) = x(1) ⊗ x(2) X h ad x = h(1) xS(h(2) ) X χ(x) = x(2) ⊗ S −1 (x(1) ). Let A and B be two YD H-module algebras. We may define a braided product, ˜ still denoted #, on the YD H-module A⊗B: X (a#b)(c#d) = ac(0) #(c(1) · b)d (1–1) for a, c ∈ A and b, d ∈ B. The braided product # makes A#B a left H-module algebra and a right H op -comodule algebra so that A#B is a YD H-module algebra. Note that the braided product # is associative. Now let A be a YD H-module algebra. The H-opposite algebra A of A is the YD H-module algebra defined as follows: A equals A as a YD H-module, with multiplication given by the formula X a◦b= b(0) (b(1) · a) 442 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG for all a, b ∈ A. In case the antipode of H is of order two, A is equal to A as a YD H-module algebra. Let M be a YD H-module such that M is of finite type. The endomorphism algebra Endk (M ) is a YD H-module algebra with the H-structures induced by those of M , i.e., for h ∈ H, f ∈ Endk (M ) and m ∈ M , P (h · f )(m) = h(1) · f (S(h(2) ) · m), (1–2) P χ(f )(m) = f (m(0) )(0) ⊗ S −1 (m(1) )f (m(0) )(1) . Recall from [11, 4.2] that the H-opposite of Endk (M ) is isomorphic as an YD H-algebra to Endk (M )op , where the latter has YD H-module structure given by P (h · f )(m) = h(2) · f (S −1 (h(1) ) · m), (1–3) P χ(f )(m) = f (m(0) )(0) ⊗ f (m(0) )(1) S(m(0) ) for m ∈ M, h ∈ H and f ∈ Endk (M ). 1.3. Quasitriangular and coquasitriangular Hopf algebras. A quasitriangular Hopf algebra is a pair (H, R), where H is a Hopf algebra with an P (1) invertible element R = R ⊗ R(2) ∈ H ⊗ H satisfying the following axioms (r = R): P P (1) (QT1) ∆(R(1) ) ⊗ R(2) = R ⊗ r(1) ⊗ R(2) r(2) , P (QT2) ε(R(1) )R(2) = 1, P (1) P (QT3) R ⊗ ∆(R(2) ) = R(1) r(1) ⊗ r(2) ⊗ R(2) , P (1) (QT4) R ε(R(2) ) = 1, cop (QT5) ∆ (h)R = R∆(h), where ∆cop = τ ∆ is the comultiplication of the Hopf algebra H cop and τ is the switch map. Now let M be a left H-module. It is well-known that there is an induced H-comodule structure on M as follows: X χ(m) = R(2) · m ⊗ R(1) (1–4) for m ∈ M such that the left H-module M together with (1–4) is a YD Hmodule. When M is a left H-module algebra, then (1–4) makes M into a right H op -comodule algebra and hence a YD H-module algebra. It is easy to see that HomH (M, N ) = HomH H (M, N ) for any two YD H-modules M, N with comodule structures (1–4) stemming from the left module structures. Thus the category H H M of left H-modules and H-morphisms can be embedded into the category Q R R as a full subcategory, which we denote by H M . Moreover H M is a braided monoidal subcategory of QH since the tensor product is closed in H M and the braiding Ψ of QH restricts to the braiding of H MR which is nothing but ΨR induced by the R-matrix: X ΨR (m ⊗ n) = R(2) · n ⊗ R(1) · m THE BRAUER GROUP OF A HOPF ALGEBRA 443 where m ∈ M, n ∈ N and M, N ∈ H MR . A coquasitriangular Hopf algebra is a pair (H, R), where H is a Hopf algebra and R ∈ (H ⊗ H)∗ is a convolution invertible element and satisfies the following axioms: (CQT1) (CQT2) (CQT3) (CQT4) R(h ⊗ 1) = R(1 ⊗ h) = ε(h)1H , P R(ab ⊗ c) = R(a ⊗ c(1) )R(b ⊗ c(2) ), P R(a ⊗ bc) = R(a(1) ⊗ c)R(a(2) ⊗ b), P P b(1) a(1) R(a(2) ⊗ b(2) ) = R(a(1) ⊗ b(1) )a(2) b(2) . Let M be a right H-comodule. There is an induced left H-module structure on M given by X h ⊲1 a = a(0) R(h ⊗ a(1) ) (1–5) for all a ∈ A, h ∈ H, such that M is a YD H-module. The right H-comodule category MH can be embedded into QH as a full braided monoidal subcategory. H We denote by MH R the braided subcategory of Q . op It is easy to check that an H -comodule algebra A with the H-module structure described in (1–5) is a YD H-module algebra. 2. Quaternion Algebras Let k be a field. Quaternion algebras play a very important role in the study of the Brauer group Br(k) of k. On the other hand, quaternion algebras also represent elements in the Brauer–Wall group BW(k) of Z 2 -graded algebras. The natural Z 2 -gradings of quaternion algebras are obtained from certain involutions related to the canonical quadratic forms of quaternion algebras. However, one may find that the same quaternion algebra will represent two different elements in BW(k). When one turns to the Brauer–Long group BD(k, Z 2 ) of Z 2 -dimodule algebras where actions of Z 2 commute with the Z 2 -gradings, the quaternion algebras now represent four different elements of order two. Now if we add a differential on the Z 2 -graded algebras such that they become differential superalgebras, we may form the Brauer group of differential superalgebras, and the quaternion algebras are now differential superalgebras. If we mimic the process used by C.T.C. Wall, we obtain a Brauer group BDS(k) of differential superalgebras. A new interesting fact now shows, i.e., a quaternion algebra may represent an element of infinite order in BDS(k). As a consequence, the Brauer group BDS(k) is a non-torsion infinite group if k has characteristic zero. Recall the definition of a quaternion algebra. For α, β ∈ k • = k\0, define a 4-dimensional algebra with basis {1, u, v, w} by the following multiplication table: uv = w, u2 = α1, v 2 = β1, vu = −w. ¡ ¢ Here 1 denotes the unit. We denote this algebra by α,β k . The elements in the subspace ku + kv + kw are called pure quaternions. The subspace of pure 444 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG quaternions is independent ¡ of¢ the choice of standard basis and is determined by the algebra structure of α,β k . There exists a canonical linear involution given by − : ³ α, β ´ k −→ ³ α, β ´ k , x = x0 + x1 = x0 − x1 ¡ ¢ where x0 ∈ k and x1 ∈ ku + kv + kw. It follows that α,β is isomorphic to its k ¡ ¢ ¡ α,β ¢op . One may easily calculate that the center of α,β is k opposite algebra k k ¡ α,β ¢ and that k has no proper ideals except {0}. An algebra is called a central simple algebra if its center is canonically isomorphic to k and it has no proper non-zero ideals. Any n×n-matrix algebra Mn (k) is a central simple algebra. The opposite algebra of a central simple algebra is obviously a central simple algebra. The tensor product of two central simple algebras is still a central simple algebra. There are several characterizations of a central simple algebra [19; 39]: Proposition 2.1. Let A be a finite dimensional algebra over a field k. The following are equivalent: (1) A is a central simple algebra. (2) A is a central separable algebra (here A is separable if mult : A ⊗ A −→ A splits as an A-bimodule map). (3) A is isomorphic to a matrix algebra Mn (D) over a skew field D where the center of D is k. (4) The canonical linear algebra map can : A⊗Aop −→ End(A) given by can(a⊗ b)(c) = acb for a, b, c ∈ A is an isomorphism. A finite dimensional algebra satisfying one of the above equivalent conditions is called an Azumaya algebra. Let B(k) be the set of all isomorphism classes of Azumaya algebras. Then B(k) is a semigroup with the multiplication induced by the tensor product and with the unit represented by the one dimensional algebra k. Define an equivalence relation ∼ on B(k) as follows: Two central simple algebras A and B are equivalent, denoted A ∼ B, if there are two positive integers m and n such that A ⊗ Mn (k) ∼ = B ⊗ Mm (k) as algebras. Then the quotient set of B(k) modulo the equivalence relation ∼ is a group and is called the Brauer group of k, denoted Br(k). The Brauer group Br(k) can be defined more intuitively as the quotient B(k)/M (k), where M (k) is a sub-semigroup generated by the isomorphism classes of matrix algebras over k. If [A] is an element in Br(k) represented by a central simple algebra A, then the inverse [A]−1 is represented by the opposite algebra Aop because A ⊗ Aop is isomorphic to a matrix algebra. The Brauer group Br(k) can be generalized to the Brauer group of a commutative ring by THE BRAUER GROUP OF A HOPF ALGEBRA 445 making use of the equivalent condition (2) or (4) of Proposition 2.1. That is, an Azumaya algebra A over a commutative ring is a faithfully projective algebra such that the condition (2) or (4) of Proposition.2.1 holds. One may refer to [2; 19] for the details on the Brauer group of a commutative ring. However, in this section we restrict our attention to the case where k is a field. Let us return to the consideration of quaternion algebras. We know that a quaternion algebra is a central simple algebra and it is isomorphic to its opposite ¡ ¢ algebra due to the canonical involution map. Thus [ α,β ] is an element of k order not greater than two. Actually any element of order two in Br(k) can be represented by a tensor product ¡ ¢ of quaternion algebras (see [39]). The quaternion algebra α,β has a canonical Z 2 -grading defined as follows: k ³ α, β ´ k = A0 + A1 , A0 = k + kw, A1 = ku + kv. (2–1) In [57], Wall introduced the notion of a Z 2 -graded Azumaya algebra which is a graded central and a graded separable algebra A in the following sense: i. the graded center Zg (A) = {a ∈ A | ab = ba0 + b0 a1 − b1 a1 , ∀b ∈ A} = k. ii. A is a simple graded algebra, i.e., A has no proper non-zero graded ideals. As in Proposition 2.1, we may replace ‘graded simplicity’ by ‘graded separability’ if the characteristic of k is different from 2. That is, condition ii can be replaced by iii. A ⊗ A −→ A splits as a Z 2 -graded A-bimodule map, where the grading on A ⊗ A is the diagonal one. ˆ of two graded algebras Given two graded algebras A and B. The product A⊗B A and B is defined as follows: ˆ ˆ = (−1)∂(b)∂(c) ac⊗bd ˆ (a⊗b)(c ⊗d) (2–2) where b and c are homogeneous elements and ∂(b), ∂(c) are the graded degrees of b and c respectively. If A and B are graded Azumaya algebras, then the product ˆ is a graded Azumaya algebra. Now one may repeat the definition of Br(k) A⊗B by adding the term ‘(Z 2 −) graded’ to obtain the Brauer group of graded algebras which is referred to as the Brauer–Wall group, denoted BW(k). Notice that in the definition of the equivalence ∼, the grading of any matrix algebra Mn (k) must be ‘good’, namely, Mn (k) ∼ = End(M ) as graded algebras for some n-dimensional graded module M . The Brauer–Wall group BW(k) can be completely described in terms of the usual Brauer group Br(k) and the group of graded quadratic extensions: 1 −→ Br(k) −→ BW(k) −→ Q2 (k) −→ 1 where Q2 (k) = Z 2 × k • /k • 2 with multiplication given by ′ (e, d)(e′ , d′ ) = (e + e′ , (−1)ee dd′ ) 446 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG for e, e′ ∈ Z 2 and d, d′ ∈ k • /k • 2 . One may write down the multiplication rule for the product Br(k) × Q2 (k) so that BW(k) is isomorphic to Br(k) × Q2 (k) (for details see [18; 46]). ¡ ¢ √ • Again let us look at the quaternion algebras α,β k , α, β ∈ k . Let kh αi be √ the graded algebra k ⊕ ku and kh βi be the graded algebra ¡ k⊕ ¢ kv. It is easy to √ show that kh αi is a graded Azumaya algebra and that α,β is isomorphic to ¡ ¢ k √ √ ˆ the graded product kh αi⊗kh βi. It follows that α,β is a graded Azumaya k ¡ α,β ¢ ­ α,β ® algebra. We denote by k the graded Azumaya algebra k in order to ­ ® ¡ ¢ √ make the difference between α,β and α,β k k . Since¡ [kh ¢ αi] ∈ BW(k) is an element of order two (or one) if α 6∈ k • 2 (or α ∈ k • 2 ), [ α,β k ] is of order equal or ¡ ¢ ­ α,β ® less than two. Though α,β and are the same algebra, they do represent k k ¡ ¢ two different elements of order two in BW(k) when α,β is a division algebra. k Furthermore, the quaternion algebras are no longer the ‘smallest’ nontrivial graded Azumaya algebras in terms of dimension. Here the smallest ones are quadratic extensions of k. This prompts the idea that the more extra structures you put on algebras, the more classes of such structured Azumaya algebras you will get, and the richer the corresponding Brauer group will be. So we look again at quadratic extensions and quaternion algebras. For a quadratic extension √ kh αi, there is a natural k-linear Z 2 -action on it: σ(1) = 1, σ(u) = −u (2–3) √ where σ is the generator of the group Z 2 and u is the generator of the field kh αi. It is easy to see that the action (2–3) commutes with the canonical ¡ ¢ grading on √ kh αi. The action (2–3) extends to any quaternion algebra α,β in the way of k diagonal group action. In fact, to any graded algebra A = A0 ⊕ A1 , one may associate a natural Z 2 -action on A as follows: σ(ai ) = (−1)i ai (2–4) where ai ∈ Ai is a homogeneous element of A. A Z 2 -graded algebra with a Z 2 -action that commutes with the grading is a Z 2 -dimodule algebra. The notion of a dimodule algebra for a finite abelian group was introduced by F. W. Long in 1972 [29], it is extended for a commutative and cocommutative Hopf algebra in [30]. Let A be a Z 2 -graded algebra. Having the canonical Z 2 -action (2–4), A is a Z 2 -dimodule algebra. The product (2–2) respects the action (2–4). In this case, we may forget the action (2–4). However, if we take any two Z 2 -dimodule algebras A and B, the graded product (2–2) may not respect actions of Z 2 . For instance, A is a graded Azumaya algebra with the action (2–4) and B is a graded Azumaya algebra with the trivial Z 2 -action (i.e., ˆ is σ acts as the identity map). Both A and B are dimodule algebras, but A⊗B not a dimodule algebra. In order to have a product for dimodule algebras, we have to modify the product (2–2) such that the action of Z 2 is involved. This is the situation dealt THE BRAUER GROUP OF A HOPF ALGEBRA 447 with by F.W. Long. Let A and B be two dimodule algebras. Long defined a product # on A ⊗ B as follows: (a#b)(c#d) = ac#σ ∂(c) (b)d (2–5) where c is a homogeneous element. The product (2–5) preserves the dimodule structures, and restricts to the product (2–2) when the dimodule algebras have the canonical action (2–4). With this product (2–5) Long was able to define the Brauer group of dimodule algebras which is now referred to as the Brauer– Long group of Z 2 and is denoted BD(k, Z 2 ). The definition of an Azumaya Z 2 -dimodule algebra is similar to the definition of a graded Azumaya algebra. Suppose that the characteristic of the field k is different from two. A Z 2 dimodule algebra is called an Azumaya dimodule algebra if A satisfies the following two conditions: i. A is Z 2 -central, namely, {a ∈ A | ab = bσ i (a), ∀b ∈ Ai } = {a ∈ A | ba = a0 b + a1 σ(b), ∀b ∈ A} = k. ii. the multiplication map A#A −→ A splits as A-bimodule and Z 2 -dimodule map. Note that the foregoing definition is not the original definition given by Long, but it is equivalent to that if the characteristic of k is different from two. The equivalence relation ∼ is defined as follows: for two Azumaya dimodule algebras A and B, A ∼ B if and only if there exists two finite dimensional dimodules M and N such that A#End(M ) ∼ = B#End(N ) as dimodule algebras. The Brauer–Long group BD(k, Z 2 ) contains the Brauer Wall-group BW(k) as a subgroup. Let ¡ α,β ¢ us investigate the role played by quaternion algebras in BD(k, Z 2 ). If k¡ is¢a quaternion algebra, then there are eight types of dimodule structures on α,β k : (1) the trivial action and the trivial grading, (2) the trivial action and the canonical grading (2–1), (3) the canonical action (2–4) and the trivial grading, ­ ® (4) the action (2–4) and the grading (2–1), i.e., the dimodule structure of α,β k , (5) the action (2–4) and the grading A0 = k ⊕ ku, A1 = kv ⊕ kw, (6) the grading (2–1) and the action given by σ(u) = u, σ(v) = −v. If we switch the roles ¡ of ¢u and v in (5) and (6), we will obtain two more dimodule structures on α,β take a while to check that the first four k . One may ¡ ¢ types of dimodule structures make α,β into Z 2 -Azumaya dimodule algebras. k ¡ ¢ ¡ α,β ¢ However, though k is an Azumaya algebra the dimodule algebra α,β of k type five or six is not a Z 2 -Azumaya algebra because it is not Z 2 -central. For instance, the left center {a ∈ A | ab = bσ i (a), ∀b ∈ Ai } = k ⊕ ku. 448 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG is not trivial in the case ¡ of¢ type five. of type (1)–(4) represent four different elements of Nevertheless, the α,β k ¡ ¢ ¡ ¢ is a division algebra. Let α,β order two in BD(k, Z 2 ) when α,β k k i be the Z 2 -Azumaya dimodule algebra of type (i), where i = 1, 2, 3, 4.¡ Since ¢ the multiplication of the group is induced by the braided product (2–5), [ α,β ] commutes ¡ ¢ ¡ α,β ¢ ¡ α,β ¢ ¡ α,β ¢ ¡ α,β ¢ k 1 with [ α,β ], but in general [ ][ ] = 6 [ ][ ] (pending on k, k i k 2 k 3 k 3 k 2 see [18, Thm]). For example, when k = R, the real number field, we have [H 2 ][H 3 ] = (1, −1, 1, −1)(1, 1, −1, −1) = (1, −1, −1, 1), [H 3 ][H 2 ] = (1, 1, −1, −1)(1, −1, 1, −1) = (1, −1, −1, −1), ¡ −1,−1 ¢ and BD(R, Z 2 ) = Z 2 × Z 2 × Z 2 × Br(R) (Br(R) = Z 2 ) with where H = k multiplication rules given by [18, Thm] or [7, 13.12.14]. In fact, BD(R, Z 2 ) ∼ = D8 , the dihedral group of 16 elements and BW(R) ∼ = Z 8 , the cyclic group of 8 elements (see [29] or [7, 13.12.15]). Thus BD(k, Z 2 ) may not be an abelian group though BW(k) is an abelian group. Nonetheless, both BW(k) and BD(k, Z 2 ) are torsion groups (see [18]). This will not be the case for the Brauer group of differential superalgebras introduced hereafter. For convenience we call a Z 2 -graded algebra a super algebra. Let A = A0 ⊕A1 be a superalgebra. A linear endomorphism δ of A is called a super-derivation of A if δ is a degree one graded endomorphism and satisfies the following condition: δ(ab) = aδ(b) + (−1)∂(b) δ(a)b where b is homogeneous and a is arbitrary. A super-derivation δ is called a differential if δ 2 = 0. A graded algebra A with a differential δ is called a differential superalgebra (simply DS algebra), denoted (A, δ) or just A if there is no confusion. Two DS algebras (A, δA ) and (B, δB ) can be multiplied by means of the graded product ˆ δA⊗B (2–2). So we obtain a new DS algebra (A⊗B, is given by ˆ ), where δA⊗B ˆ ˆ = a⊗δ ˆ B (b) + (−1)i δA (a)⊗b ˆ δA⊗B ˆ (a⊗b) for a ∈ A and b ∈ Bi . Let M be a graded module. M is called a differential graded module if there 2 = 0 exists a degree one graded linear endomorphism δM of M such that δM (in the sequel δM will be called a differential on M ). The endomorphism ring End(M ) is a DS algebra. The grading on End(M ) is the induced grading and the differential δ on End(M ) is induced by δM , namely, δ(f )(m) = f (δM (m)) + (−1)∂(m) δ(f (m)) (2–6) for any homogeneous element m ∈ M and f ∈ End(M ­ α,β ® ). Now let A be the graded Azumaya algebra k . There is a natural differential on A given by Doi and Takeuchi [22]: δ(1) = δ(u) = 0, δ(v) = 1, δ(w) = u. (2–7) THE BRAUER GROUP OF A HOPF ALGEBRA 449 So any quaternion­ algebra is a DS algebra. As mentioned before, the graded ® represents an element of order not greater than two in Azumaya algebra α,β k ˆ is a graded BW(k). This means that the product graded Azumaya algebra A⊗A 4 × 4-matrix algebra. So there is a 4-dimensional graded module M such that ˆ ∼ A⊗A = End(M ) as graded ­ ® algebras. Is this the same when we add the canonical differential (2–7) to α,β k ? In other words, does there exist a graded module M ˆ is isomorphic to End(M ) as a DS algebra? with a differential δM such that A⊗A ® ­ . Before we The answer is even negative for the graded matrix algebra 1,−1 k answer the question let us first define the Brauer group of DS algebras. Definition 2.2. A DS algebra A is called a DS Azumaya algebra if A is a graded Azumaya algebra. Two DS Azumaya algebras A and B are said to be equivalent, denoted A ∼ B, if there are two differential graded module M and ˆ ˆ N such that A⊗End(M )∼ ) as DS algebras. = B ⊗End(N Let B(k) be the set of isomorphism classes of DS Azumaya algebras. It is a routine verification that the quotient set of B(k) modulo the equivalence relation ∼ is a group and is called the Brauer group of DS algebras, denoted BDS(k). If A represents an element [A] of BDS(k), then the graded opposite A represents the inverse of [A] in BDS(k), where A and A share the same differential. The unit of the group BDS(k) is represented by matrix DS algebras which are the endomorphism algebras of some finite dimensional differential graded modules. In other words, if A is a DS Azumaya algebra such that [A] = 1, then A is isomorphic to End(M ) as a DS algebra for some finite dimensional differential graded module M . This follows from the fact that the Brauer equivalence ∼ is the same as the differential graded Morita equivalence which can be done straightforward by adding the ‘differential’ to graded Morita equivalence. We would rather wait till next section to see a far more general ­ α,β ® H-Morita• theory. Following Definition 2.2 all quaternion algebras k , α, β ∈ k , are DS ­ ® Azumaya algebras. We show first that the DS matrix algebra α,−α does not k represents the unit of BDS(k). In order to prove this, we need to consider the associated automorphism σ given by (2–4) of a differential graded module M . Since the differential δ of M is a degree one graded endomorphism, it follows that δ anti-commutes with σ, namely, σδ + δσ = 0. ­ ® Lemma 2.3. For any α ∈ k • , the DS algebra [ α,−α ] 6= 1 in BDS(k). k ® ­ ® ­ and assume that [ α,−α ] = 1 in BDS(k). Then Proof. Let A be α,−α k k ­ ® ∼ there exists a two dimensional differential graded module M such that α,−α = k End(M ). Since the differential δE and the automorphism σE of End(M ) are induced by the differential δM and automorphism σM of M respectively (see (2–6) for the differential), A ∼ = End(M ) implies that there exist two elements ν and ω in A such that the canonical differential δ given by (2–7) is induced by ν and the automorphism σA given by (2–4) is the inner automorphism induced by 450 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG ω, i.e., δ(a) = aν − (−1)∂(a) νa, σ(b) = ωbω −1 where a, b ∈ A and a is homogeneous. Furthermore, ν and ω satisfy the relations that δM and σM obey, i.e., ν 2 = 0, ω 2 = 1, νω + ων = 0. ® ­ . Then we have the following relations: Let u, v be the two generators of α,−α k δ(u) = uν + νu = 0, δ(v) = vν + νv = 1, σ(u) = ωuω −1 = −u, σ(v) = ωvω −1 = −v. −1 It follows that ν = − α2 v + suv for some s ∈ k and ω = α−1 uv. Since ν 2 = 0, we have α−1 α−1 0 = (− v + suv)2 = − + s2 α2 2 4 So s cannot be zero. However, the anti-commutativity of ν with ω implies that 0 = νω + ων = (− 21 α−1 v + suv)α−1 uv + α−1 uv(− 21 α−1 + suv) v = 2sα. So s must ­ α,−αbe ® zero. Contradiction! Thus we have proved that it is impossible to ∼ have ) for some 2-dimensional differential graded module M , k ­ = End(M ® α,−α ¤ and hence [ k ] 6= 1. ­ α,−α ® From Lemma 2.3 we see that a DS matrix algebra (α ∈ k • ) representing k the unit in BW(k) ® represents a non-unit element in BDS(k). In the following ­ now represents an element of infinite order in BDS(k) if the we show that α,−α k characteristic of k is zero. In fact: Proposition 2.4 [54, Prop. 7]. Let (k, +) be the additive group of k. Then τ : (k, +) −→ BDS(k), α 7→ ·D −1 E¸ α , −α−1 , k α 6= 0, 0 7→ 1 is a group monomorphism. Proof. By Lemma 2.3 it is sufficient to show that τ is a group homomor­ −1 −1 ,0 ® ­ β −1 ,−β −1 ,0 ® ˆ ⊗ . If α + β = 0, then phism. Consider the product α ,−α k k ­ β −1 ,−β −1 ,0 ® ­ α−1 ,−α−1 ,0 ® = and k k D α−1 , −α−1 , 0 E D β −1 , −β −1 , 0 E ∼ ˆ k ⊗ k D α−1 , −α−1 , 0 E = End k , THE BRAUER GROUP OF A HOPF ALGEBRA 451 which represents the unit in BDS(k). ­ −1 −1 ® Assume that α+β 6= 0. Let {u, v} and {u′ , v ′ } be the generators of α ,−α k ­ −1 −1 ® respectively. Let and of β ,−β k U= αβ α β ˆ ′ + w⊗u ˆ ′ ), V = ˆ + ˆ ′. (u⊗w v ⊗1 1⊗v α+β α+β α+β Then U 2 = (α + β)−1 , V 2 = −(α + β)−1 , U V + V U = 0. ¡ −1 −1 ¢ Thus U and V generate the matrix algebra σ ,−σ , where σ = α + β. One k may further check that the induced Z 2 -grading and the induced differential on ¡ σ−1 ,−σ−1 ¢ are given by (2–1) and (2–7). Thus U and V generate a DS quaterk ­ ­ −1 −1 ® ­ β −1 ,−β −1 ® −1 −1 ® ˆ nion subalgebra (α+β) ,−(α+β) ⊗ in α ,−α . Applying the k k k commutator theorem for Azumaya algebras (see [19]), we obtain D α−1 , −α−1 E D β −1 , −β −1 E ˆ ⊗ D σ −1 , −σ −1 E ⊗ M2 (k) k k k as algebras, where σ = α + β. We leave it to readers to check that they are equal as DS algebras (or see [55, Coro.2]). It follows that τ (α)τ (β) = = ·D −1 E D β −1 , −β −1 E¸ α , −α−1 ˆ k ⊗ k = ·D E¸ (α + β)−1 , −(α + β)−1 k = τ (α + β) in the Brauer group BDS(k). So we have proved that τ is a group homomorphism. ¤ Note that when the characteristic of k is 0, ­ (k, ®+) is not a torsion group. The elin BDS(k) generates a subgroup ement represented by the matrix algebra 1,−1 k ­ ® with the which is isomorphic to Z. In this case any quaternion algebra α,β k canonical grading and the canonical differential represents an element of infinite ­ ® order in the Brauer group BDS(k). If the characteristic of k is p 6= 2, then α,β k represents an element of order not greater than p in BDS(k). The group (k, +) indicates the substantial difference between the Brauer–Wall group BW(k) and the Brauer group BDS(k) of DS algebras. Actually, this subgroup comes only from extra differentials added to graded Azumaya algebras. Theorem 2.5 [54, Thm. 8]. BDS(k) = BW(k) × (k, +). Proof. By definition of a DS Azumaya algebra, we have a well-defined group homomorphism γ : BDS(k) −→ BW(k), [A] −→ [A] by forgetting the differential on the latter A. It is clear that γ is a surjective map as a graded Azumaya algebra with a trivial differential is a DS Azumaya algebra. ­ −1 −1 ® represents the unit in BW(k), we Since the graded Azumaya algebra α ,−α k have τ (k, +) ⊆ Ker(γ). To prove that Ker(γ) ⊆ τ (k, +), we need to use the 452 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG associated automorphism σ given by (2–4) of a graded algebra. Let A be a DS Azumaya algebra representing a non-trivial element in Ker(γ). Since [A] = 1 in BW(k), A is a graded matrix algebra. Since A is Azumaya, the associated automorphism σ is an inner automorphism induced by some invertible element u ∈ A such that u2 = 1. Similarly, the differential δ is an inner super-derivation induced by some element v ∈ A in the sense that δ(a) = va − (−1)∂(a) va for any homogeneous element a ∈ A. Note that v 2 6= 0 by the proof of Lemma 2.3. 2 Now one may apply the properties that δ 2 = 0, σδ + δσ obtain ¡ α,β ¢ = 0 and σ = 1 to that u and v generate a quaternion subalgebra k for some α, β ∈ k • with α ¢ a square number. Here u, v are not necessarily ¡ α,β ¢ the canonical generators of ¡being α,β (see [54, Thm. 8] for more detail). Thus is a matrix algebra and A is k k ¡ ¢ a tensor product α,β ⊗ M (k) of two matrix algebras for some integer n. Since n k ¢ ¡ α,β ¡ α,β ¢ u and v generate k and Mn (k) commutes with k , σ and δ act on Mn (k) ¡ ¢ ¡ ¢ ˆ trivially and α,β is a DS subalgebra of A. It follows that A = α,β k k ⊗Mn (k). Finally ¡one ¢may take a while to check¡that¢ there is a pair of ­new® generators such that the DS algebra α,β be written as α,β with u′ , v ′ u′ , v ′ of α,β k k ­ can ® ­ α,βk ® α,β ˆ being the canonical generators. So [A] = [ k ⊗Mn (k)] = [ k ] ∈ τ (k, +). Finally, since γ is split by the inclusion map, the Brauer group BDS(k) is a direct product of BW(k) with (k, +). ¤ DS algebras may be generalized to differential Z 2 -dimodule algebras adding one differential to a dimodule algebra such that the action of the differential anticommutes with the action of the non-unit element of Z 2 . The Brauer group BDD(k, Z 2 ) of differential dimodule algebras can be defined and computed. Once again quaternion algebras play the same roles as they do in the Brauer group of DS algebras. As an exercise for readers, the Brauer group BDD(k, Z 2 ) is isomorphic to the group (k, +) × BD(k, Z 2 ) [56]. Other exercises include adding more differentials, say n differentials δ1 , · · · , δn , to graded Azumaya algebras or dimodule Azumaya algebras. For instance, one may obtain the Brauer group BDSn (k) of n-differential superalgebras which is isomorphic to the group (k, +)n ×BW(k). From the proofs of Lemma 2.3 and Theorem 2.5 one may find that the argument there is actually involved with actions of an automorphism and a differential which satisfy the relations: σ 2 = 1, δ 2 = 0, σδ + δσ = 0 where σ is the non-unit element of Z 2 . In fact, the four dimensional algebra generated by σ and δ is a Hopf algebra with comultiplication given by ∆(σ) = σ ⊗ σ, ∆(δ) = 1 ⊗ δ + δ ⊗ σ THE BRAUER GROUP OF A HOPF ALGEBRA 453 and counit given by ε(σ) = 1 and ε(δ) = 0. This Hopf algebra is called Sweedler Hopf algebra, denoted H4 . The two generators σ and δ are usually replaced by g and h. Thus a DS algebra is nothing else but an H4 -module algebra. Conversely if A is an H4 -module algebra, there is a natural Z 2 -grading on A given by A0 = {x ∈ A | g(x) = x}, A1 = {x ∈ A | g(x) = −x}. (2–8) With respect to the grading (2–8), the action of h is a differential on A so that A is a DS algebra. Moreover, the H4 -module algebra is a YD H4 -module algebra with the coaction given by the grading (2–8). In this way we may identify DS algebras with YD H4 -module algebras with coactions given by the grading (2–8). This is the reason why the Brauer group of DS algebras can be defined. The elements in the Brauer group of DS algebras are eventually represented by those so called YD H4 -Azumaya algebras which will be introduced in the next section. In particular, quaternion algebras are YD H4 -Azumaya algebras. 3. The Definition of the Brauer Group Throughout this section H is a flat k-Hopf algebra with a bijective antipode S, and all k-modules (except H) are faithfully projective over k. Let A be a YD H-module algebra. The two YD H-module algebras A#A and A#A are called the left and right H-enveloping algebras of A (see (1–1) for definition of #). We are now able to define the concept of an H-Azumaya algebra, and construct the Brauer group of the Hopf algebra H. Definition 3.1. A YD H-module algebra A is called an H-Azumaya algebra if it is faithfully projective as a k-module and if the following YD H-module algebra maps are isomorphisms: P ˆ F : A#A −→ End(A), F (a⊗b)(x) = ax(0) (x(1) · b), P G : A#A −→ End(A)op , G(a#b)(x) = a(0) (a(1) · x)b. where the YD H-structures of End(A) and End(A)op are given by (1–2) and (1–3). It follows from the definition that a usual Azumaya algebra with trivial Hstructures is an H-Azumaya algebra. One may take a while to check (or see [11]) that the H-opposite algebra A of an H-Azumaya algebra A and the braided product A#B of two H-Azumaya algebras A and B are H-Azumaya algebras. In particular, the YD H-module algebra End(M ) of any faithfully projective YD H-module M is an H-Azumaya algebra. An H-Azumaya algebra of the form End(M ) is called an elementary H-Azumaya algebra. As usual we may define an equivalence relation on the set B(k, H) of isomorphism classes of H-Azumaya algebras. 454 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG Definition 3.2. Let A and B be two H-Azumaya algebras. A and B are said to be Brauer equivalent, denoted A ∼ B, if there exist two faithfully projective YD H-modules M and N such that A#End(M ) ∼ = B#End(N ) as YD H-module algebras. As expected the quotient set of B(k, H) modulo the Brauer equivalence is a group with multiplication induced by the braided product # and with inverse operator induced by the H-opposite ¯ . Denote by BQ(k, H) the group B(k, H)/ ∼ and call it the Brauer group of the Hopf algebra H or the Brauer group of Yetter– Drinfel’d H-module algebras. Since a usual Azumaya algebra with trivial YD H-module structures is H-Azumaya and the Brauer equivalence restricts to the usual Brauer equivalence, the classical Brauer group Br(k) of k is a subgroup of BQ(k, H) sitting in the center of BQ(k, H). Let E be a commutative ring with unit. Suppose that we have a ring homomorphism f : k −→ E. By usual base change HE = H ⊗k E is a E-Hopf algebra. Now in a way similar to [30, 4.7, 4.8] we obtain an induced group homomorphism on the Brauer group level. Proposition 3.3. The functor M 7→ M ⊗k E induces a group homomorphism BQ(k, H) −→ BQ(E, HE ), mapping the class of A to the class of AE . The kernel of the foregoing homomorphism, denoted by BQ(E/k, H), is called the relative Brauer group of H w.r.t. the extension E/k. Denote by BQs (k, H) the union of relative Brauer groups BQ(E/k, H) of all faithfully flat extensions E of k. BQs (k, H) is called the split part of BQ(k, H). In [12], BQs (k, H) was described in a complex: 1 −→ BQs (k, H) −→ BQ(k, H) −→ O(E(H)) where O(E(H)) is a subgroup of the automorphism group Aut(E(H)) and E(H) is the group of group-like elements of the dual Drinfel’d double D(H)∗ of H (see [12, 3.11-3.14] for details). Now let H be a commutative and cocommutative Hopf algebra. In this situation, a YD H-module (algebra) is an H-dimodule (algebra). But an H-Azumaya algebra in the sense of Definition 3.1 is not an Azumaya H-dimodule algebra in the sense of Long (see [30] for detail on the Brauer group of dimodule algebras we refer to [6; 30]). The reason for this is that the braided product we choose in QH is the inverse product of DH when H is commutative and cocommutative. However we have the following: Proposition 3.4 [12, Prop.5.8]. Let H be a commutative and cocommutative Hopf algebra. If A is an H-Azumaya algebra, then Aop is an H-Azumaya dimodule algebra. Moreover , BQ(k, H) is isomorphic to BD(k, H). The isomorphism is given by [A] 7→ [Aop ]−1 . THE BRAUER GROUP OF A HOPF ALGEBRA 455 When H is a commutative and cocommutative Hopf algebra, the Brauer–Long group BD(k, H) has two subgroups BM(k, H) and BC(k, H) (see [30, 1.10, 2.13]). The subgroup BM(k, H) consists of isomorphism classes represented by H-Azumaya dimodule algebras with trivial H-comodule structures, and the subgroup BC(k, H) consists of isomorphism classes represented by H-Azumaya dimodule algebras with trivial H-module structures. These two subgroups were calculated by M. Beattie in [3] which are completely determined by the groups of Galois objects of H and H ∗ respectively (e.g., see Corollary 4.3.5). If H is not cocommutative (or not commutative) an H-module (or comodule) algebra with the trivial H op -comodule (or the trivial H-module) structure does not need to be YD H-module algebra. In general, we do not have subgroups like BM(k, H) or BC(k, H) in the Brauer group BQ(k, H) when H is non-commutative or non-cocommutative. However, when H is quasitriangular or coquasitriangular, we have subgroups similar to BM(k, H) or BC(k, H). It is well known that the notion of a quasitriangular Hopf algebra is the generalization of the notion of a cocommutative Hopf algebra. If (H, R) is a quasitriangular Hopf algebra, an H-module algebra is automatically a YD Hmodule algebra with the freely granted H op -comodule structure (1–4). Since R H and the braided product # given by (1–1) H M is a braided subcategory of Q commutes (or is compatible) with the H op -coaction (1–4), the canonical Hlinear map F and G in Definition 3.1 are automatically H op -colinear. Thus the subset of BQ(k, H) consisting of isomorphism classes represented by H-Azumaya algebras with H op -coactions of the form (1–4) stemming from H-actions is a subgroup, denoted BM(k, H, R). We now have the following inclusions for a QT Hopf algebra. Br(k) ⊆ BM(k, H, R) ⊆ BQ(k, H). Similarly if (H, R) is a coquasitriangular Hopf algebra, the Brauer group BQ(k, H) possesses a subgroup BC(k, H, R) consisting of isomorphism classes represented by H-Azumaya algebras with H-actions (1–5) stemming from the H op -coactions. In this case we have the following inclusions of groups for a CQT Hopf algebra: Br(k) ⊆ BC(k, H, R) ⊆ BQ(k, H) When H is a finite commutative and cocommutative Hopf algebra, a CQT structure can be interpreted by a Hopf algebra map from H into H ∗ . As a matter of fact, there is one-to-one correspondence between the CQT structures on H and the Hopf algebra maps from H to H ∗ which form a group Hopf(H, H ∗ ) with the convolution product. The correspondence is given by {CQT structures on H} −→ Hopf(H, H ∗ ), R 7→ θR , θR (h)(l) = R(h ⊗ l) for any h, l ∈ H. In this case the Brauer group BC(k, H, R) is Orzech’s Brauer group Bθ (k, H) of BD(k, H) consisting of classes of θ-dimodule algebras. For θ-dimodule algebras one may refer to [36] in the case that H is a group Hopf 456 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG algebra with a finite abelian group and to [6, § 12.4] in the general case. In a special case that H = kG is a finite abelian group algebra, γ : G × G −→ k • a bilinear map, we may view γ as a coquasitriangular structure on H. Then the Brauer group Bγ (k, G) of graded algebras investigated by Childs, Garfinkel and Orzech (see [14; 15]) is isomorphic to BC(k, H, γ). Note that a cocommutative Hopf algebra with a coquasitriangular structure is necessarily commutative. Similarly, a commutative quasitriangular Hopf algebra is cocommutative. Now let us consider the finite case. Suppose that H is a faithfully projective Hopf algebra. Then the Drinfel’d double D(H) is a quasitriangular Hopf algebra with the canonical QT structure R represented by a pair of dual bases of H and H ∗ , e.g., [23; 40], and there is a one-to-one correspondence between left D(H)module algebras and Yetter–Drinfel’d H-module algebras, [32]. It follows that BQ(k, H) = BM(k, D(H), R). So BQ(k, H) ⊆ BQ(k, D(H)). Now write Dn (H) for D(Dn−1 (H)), the n-th Drinfel’d double. Then we have the following chain of inclusions: BQ(k, H) ⊆ BQ(k, D(H)) ⊆ BQ(k, D2 (H)) ⊆ · · · ⊆ BQ(k, Dn (H)) ⊆ · · · A natural question arises: when is the foregoing ascending chain finite? To end this section, let us look once again at Definition 3.1 and the definition of the Brauer equivalence. It is not surprising that these definitions are essentially categorical in nature. This means that an H-Azumaya algebra can be characterized in terms of monoidal category equivalences. The Brauer equivalence is in essence the Morita equivalence. In particular, the unit in the Brauer group BQ(k, H) is only represented by elementary H-Azumaya algebras. Let A be a YD H-module algebra. A left A-module M in QH is both a left A-module and a YD H-module satisfying the compatibility conditions: P i. h · (am) = (h(1) · a)(h(2) · m), P ii. χ(am) = a(0) m(0) ⊗ m(1) a(1) . op That is, M is a left A#H-module and a right Hopf module in A MH . Here A#H is the usual smash product rather than the braided product. Denote by H the category of left A-modules in QH and A-module morphisms in QH . AQ Similarly, we may define a right A-module M in QH as a right A-module and a YD H-module such that the following two compatibility conditions hold: P i. h · (ma) = (h(1) · m)(h(2) · a), P ii. χ(ma) = m(0) a(0) ⊗ a(1) m(1) . H Denote by QH A the category of right A-modules in Q and their morphisms. Now let A and B be two YD H-module algebras. An (A-B)-bimodule M in QH is an H (A-B)-bimodule which belongs to A QH and QH B . Denote by A QB the category of (A-B)-bimodules. View k as a trivial YD H-module algebra. Then A QH is THE BRAUER GROUP OF A HOPF ALGEBRA 457 just the category of (A-k)-bimodules in QH . Similarly QH A is the category of (k-A)-bimodules in QH . Definition 3.5. A (strict) Morita context (A, B, P, Q, ϕ, ψ) is called a (strict) H-Morita context in QH if the following conditions hold: (1) A and B are YD H-module algebras, (2) P is an (A-B)-bimodule in QH and Q is an (B-A)-bimodule in QH , ˜ B Q −→ A and (3) ϕ and ψ are (surjective) YD H-module algebra maps: ϕ : P ⊗ ˜ ψ : Q⊗A P −→ B. An H-Morita context in QH is a usual Morita context if one forgets the Hstructures. When one works with the base category QH , the usual Morita theory applies fully and one obtains an H-Morita theory in QH . Here are few basic properties of H-Morita contexts. Proposition 3.6. (1) If P is a faithfully projective YD H-module, then (End(P ), k, P, P ∗ , ϕ, ψ) is a strict H-Morita context in QH . Here ϕ and ψ are given by ϕ(p ⊗ f )(x) = pf (x) and ψ(f ⊗ p) = f (p). (2) Let B be a YD H-module algebra. If P ∈ QH B is a B-progenerator , then (A = EndB (P ), B, P, Q = HomB (P, B), ϕ, ψ) is a strict H-Morita context in QH . Here ϕ and ψ are given by ϕ(p ⊗ f )(x) = pf (x) and ψ(f ⊗ p) = f (p), where EndB (P ) is a YD H-module algebra with adjoint H-structures given in Subsection 1.1. Like usual Morita theory, if (A, B, P, Q, ϕ, ψ) is a strict HMorita context, then the pairs of functors ˜ A− : Q⊗ H AQ ˜ B− : −→ B QH and P ⊗ H ˜ A P : QH −⊗ A −→ QB H BQ −→ A QH , H ˜ B Q : QH and − ⊗ B −→ QA define equivalences between the categories of bimodules in QH . Let A be a YD H-module algebra, A is the H-opposite of A. Write Ae for A#A and e A for A#A. Then A may be regarded as a left Ae -module and a right e A-module as follows: X X (3–1) (a#b) · x = ax(0) (x(1) · b), and x · (a#b) = a(0) (a(1) · x)b. It is clear that A with foregoing Ae and e A-module structures is in Ae QH and QHe A respectively. Now consider the categories Ae QH and QHe A . To a left Ae -module M in Ae QH we associate a YD H-submodule M A = {m ∈ M | (a#1)m = (1#a)m, ∀a ∈ A}. This correspondence gives rise to a functor (−)A from Ae QH to QH . On the ˜ from QH to Ae QH . It is easy to other hand, we have an induction functor A⊗− 458 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG see that ˜ : QH A⊗− (−)A : H Ae Q ˜ −→ Ae QH , N 7→ A⊗N, −→ QH , (3–2) M 7→ M A . is an adjoint pair of functors. Similarly we have an adjoint pair of functors between categories QH and QHe A : ˜ : QH −→ QHe A , −⊗A A (−) : QHe A −→ QH , ˜ N 7→ N ⊗A, M 7→ A (3–3) M where A M = {m ∈ M | m(1#a) = m(a#1), ∀a ∈ A}. Proposition 3.7 [12, Prop.2.6]. Let A be a YD H-module algebra. Then A is H-Azumaya if and only if (3–2) and (3–3) define equivalences of categories. In fact, (3–2) and (3–3) define the equivalences between the braided monoidal categories if A is H-Azumaya (see [12]). With the previous preparation one is able to show that the ‘Brauer equivam lence’ is equivalent to the ‘H-Morita equivalence’ in QH . We denote by A∼B that b A is H-Morita equivalent to B, and denote by A∼B that A is Brauer equivalent to B, i.e., [A] = [B] ∈ BQ(k, H). b Theorem 3.8 [12, Thm. 2.10]. Let A, B be H-Azumaya algebras. A∼B if and m only if A∼B. As a direct consequence, we have that if [A] = 1 in BQ(k, H), then A ∼ = End(P ) for some faithfully projective YD H-module P . 4. An Exact Sequence for the Brauer Group BC(k, H, R) As we explained in Section 3, when a Hopf algebra H is finite, the Brauer group BQ(k, H) of H is equal to the Brauer group BC(k, D(H)∗ , R). So in the finite case, it is sufficient to consider the Brauer group BC(k, H, R) of a finite coquasitriangular Hopf algebra. We present a general approach to the calculation of the Brauer group BC(k, H, R). The idea of this approach is basically the one of Wall in [57] where he introduced the first Brauer group of structured algebras, i.e., the Brauer group of super (or Z 2 -graded) algebras which is now called the Brauer–Wall group, denoted BW(k). He proved that the Brauer group of super algebras over a field k is an extension of the Brauer group Br(k) by the group Q2 (k) of Z 2 -graded quadratic extensions of k, i.e., there is an exact sequence of group homomorphisms: 1 ✲ Br(k) ✲ BW(k) ✲ Q2 (k) ✲ 1 In 1972, Childs, Garfinkel and Orzech studied the Brauer group of algebras graded by a finite abelian group [14], and Childs (in [15]) generalized Wall’s sequence by constructing a non-abelian group Galz(G) of bigraded Galois objects THE BRAUER GROUP OF A HOPF ALGEBRA 459 replacing the graded quadratic group of k in Wall’s sequence. They obtained an exact group sequence: 1 −→ Br(k) −→ BC(k, G, φ) −→ Galz(G) where Galz(G) may be described by another exact sequence when G is a p-group. The complexity of the group Galz(G) is evident. An object in Galz(G) involves both two-sided G-gradings and two-sided G-actions such that the actions and gradings commute. In 1992, K. Ulbrich extended the exact sequences of Childs to the case of the Brauer–Long group of a commutative and cocommutative Hopf algebra (see [48]). The technique involved is essentially the Hopf Galois theory of a finite Hopf algebra. However, when the Hopf algebra is not commutative, a similar group of Hopf Galois objects does not exists although one still obtains a group of Hopf bigalois objects with respect to the cotensor product (see [42; 55]). The idea of this section is to apply the Hopf quotient Galois theory. This requires the deformation of the Hopf algebra H. Throughout, (H, R) is a finite CQT Hopf algebra. 4.1. The algebra HR . We start with the definition of the new product ⋆ on the k-module H: X h⋆l= l(2) h(2) R(S −1 (l(3) )l(1) ⊗ h(1) ) X = h(2) l(1) R(l(2) ⊗ S(h(1) )h(3) ) where h and l are in H. (H, ⋆) is an algebra with unit 1. We denote by HR the algebra (H, ⋆). It is easy to see that the counit map ε of H is still an augmentation map from HR to k. There is a double Hopf algebra for a CQT Hopf algebra (H, R) (not necessarily finite). This double Hopf algebra, denoted D[H] due to Doi and Takeuchi (see [21]), is equal to H ⊗ H as a coalgebra with the multiplication given by X (h ⊗ l)(h′ ⊗ l′ ) = hh′(2) ⊗ l(2) l′ R(h′(1) ⊗ l(1) )R(S(h′(3) ) ⊗ l(3) ) for h, l, h′ and l′ ∈ H. The antipode of D[H] is given by S(h ⊗ l) = (1 ⊗ S(l))(S(h) ⊗ 1) for all h, l ∈ H. The counit of D[H] is ε ⊗ ε. Since H is finite, the canonical Hopf algebra homomorphism Θl : H −→ H ∗ op given by Θl (h)(l) = R(h ⊗ l) induces an Hopf algebra homomorphism from D[H] to D(H), the Drinfel’d quantum double H ∗ op ⊲⊳ H. Φ : D[H] −→ D(H), Φ(h ⊲⊳ l) = Θl (h) ⊲⊳ l. When Θl is an isomorphism, we may identify D[H] with D(H). Thus a YD H-module is automatically a left D[H]-module. Moreover, the following algebra 460 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG monomorphism φ shows that HR can be embedded into D[H]. X φ : HR −→ D[H], φ(h) = S −1 (h(2) ) ⊲⊳ h(1) . Thus we may view HR as a subalgebra of the double D[H]. Moreover, one may check that the image of φ in D[H] is a left coideal of D[H]. We obtain the following: Proposition 4.1.1. HR is a left D[H]-comodule algebra with the comodule structure given by X χ : HR −→ D[H] ⊗ HR , χ(h) = (S −1 (h(3) ) ⊲⊳ h(1) ) ⊗ h(2) . The left D[H]-comodule structure of HR in Proposition 4.1.1 demonstrates that HR can be embedded into D[H] as a left coideal subalgebra. In fact, HR can be further embedded into D(H) as a left coideal subalgebra. Corrollary 4.1.2. The composite algebra map HR φ✲ D[H] Φ✲ D(H) is injective, and HR is isomorphic to a left coideal subalgebra of D(H). Let us now consider Yetter–Drinfel’d H-modules and HR -bimodules. Let M be a Yetter–Drinfel’d module over H, or a left D(H)-module. The following composite map: φ⊗ι Φ⊗ι HR ⊗ M −→ D[H] ⊗ M −→ D(H) ⊗ M makes M into a left HR -module. If we write −⊲ for the above left action, then we have the explicit formula: X h −⊲ m = (h(2) · m(0) )R(S −1 (h(4) ) ⊗ h(3) m(1) S −1 (h(1) )) (4–1) for h ∈ HR and m ∈ M . Since there is an augmentation map ε on HR , we may define the HR -invariants of a left HR -module M which is M HR = {m ∈ M | h −⊲ m = ε(h)m, ∀h ∈ HR }. When a left HR -module comes from a YD H-module we have X m(0) R(h ⊗ m(1) ), ∀h ∈ H}. M HR = {m ∈ M | h · m = h ⊲1 m = Now we define a right HR -module structure on a YD H-module M . Observe that the right H-comodule structure of M induces two left H-module structures. The first one is (1–5), and the second one is given by X h ⊲2 m = m(0) R(S(m(1) ) ⊗ h) (4–2) for h ∈ H and m ∈ M . With this second left H-action (4–2) on M , M can be made into a right D[H]-module. THE BRAUER GROUP OF A HOPF ALGEBRA 461 Lemma 4.1.3. Let M be a Yetter–Drinfel’d H-module. Then M is a right D[H]-module defined by m ↽ (h ⊲⊳ l) = S(l) ⊲2 (S(h) · m) for h, l ∈ H and m ∈ M . Moreover , if A is a YD H-module algebra, then A is a right D[H]cop -module algebra. The right D[H]-module structure in Lemma 4.1.3 does not match the canonical left D[H]-module structure induced by the Hopf algebra map Φ so as to yield a D[H]-bimodule structure on M . However the right HR -module structure on M given by ι⊗φ M ⊗ HR −→ M ⊗ D[H] −→ M, more precisely: m ⊳− h = X (h(3) · m(0) )R(h(4) m(1) S −1 (h(2) ) ⊗ h(1) ) (4–3) for m ∈ M and h ∈ HR , together with the left HR -module structure (4–1), defines an HR -bimodule structure on M . Proposition 4.1.4. Let M be a YD H-module. Then M is an HR -bimodule via (4–1) and (4–3). If A is a YD H-module algebra, then Proposition 4.1.4 implies that A is an HR -bimodule algebra in the sense that: P h −⊲ (ab) = (h(−1) · a)(h(0) −⊲ b) P (4–4) (ab) ⊳− h = (a · h(0) )(b ⊳− h(−1) ) P for a, b ∈ A and h ∈ HR , where χ(h) = h(−1) ⊗ h(0) ∈ D[H] ⊗ HR . To end this subsection we present the dual comodule version of (4–4) which is ∗ is a left needed in the next subsection. Observe that the dual coalgebra HR ∗ ∗ D[H] -module quotient coalgebra of the dual Hopf algebra D[H] in the sense that the following coalgebra map is a surjective D[H]-comodule map: ∗ φ∗ : D[H]∗ −→ HR , p ⊲⊳ q 7→ qS −1 (p). ∗ -comodule Thus a left (or right) D[H]-comodule M is a left (or right) HR ∗ ∗ in the natural way through φ . In order to distinguish D[H] or HR -comodule structures from the H-comodule structures (e.g., a YD H-module has all three comodule structures) we use different uppercase Sweedler sigma notations: P [−1] P [0] i. x ⊗ x[0] , x ⊗ x[1] stand for left and right D[H]∗ -comodule structures, P (−1) P (0) ∗ -comodule ii. x ⊗ x(0) , x ⊗ x(1) stand for left and right HR structures, 462 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG where x is an element in a due comodule. Now let A be a YD H-module algebra. Then A is both a left and right D[H]-module algebra, and therefore an HR bimodule algebra in the sense of (4–4). Thus the dual comodule versions of the formulas in (4–4) read as follows: P P (ab)(0) ⊗ (ab)(1) = a[0] b(0) ⊗ a[1] ⇁ b(1) , (4–5) P P [−1] (ab)(−1) ⊗ (ab)(0) = b ⇁ a(−1) ⊗ a(0) b[0] ∗ for a, b ∈ A, where ⇁ is the left action of D[H]∗ on HR . We will call A a right ∗ (or left) HR -comodule algebra in the sense of (4–5). Finally, for a YD H-module M , we will write M⋄ (or ⋄ M ) for the right (or ∗ -coinvariants. For instance, left) HR X M⋄ = {m ∈ M | m(0) ⊗ m(1) = m ⊗ ε.} It is obvious that M⋄ = M HR . 4.2. The group Gal(HR ). We are going to construct a group Gal(HR ) of ‘Galois’ objects for the deformation HR . The group Gal(HR ) plays the vital role in an exact sequence to be constructed. Definition 4.2.1. Let A be a right D[H]∗ -comodule algebra. A/A⋄ is said to ∗ be a right HR -Galois extension if the linear map X ∗ , β r (a ⊗ b) = a(0) b ⊗ a(1) β r : A ⊗A⋄ A −→ A ⊗ HR is an isomorphism. Similarly, if A is a left D[H]∗ -comodule algebra, then A/⋄ A is said to be left Galois if the linear map X ∗ β l : A ⊗⋄ A A −→ HR ⊗ A, β l (a ⊗ b) = b(−1) ⊗ ab(0) is an isomorphism. If in addition the subalgebra ⋄ A (or A⋄ ) is trivial, then A is ∗ -Galois object. For more detail on Hopf quotient Galois called a left (or right) HR theory, readers may refer to [33; 44; 45]. ∗ The objects we are interested in are those HR -bigalois objects which are both ∗ ∗ -coactions commute. left and right HR -Galois such that the left and right HR ∗ Denote by E(HR ) the category of YD H-module algebras which are HR -bigalois objects. The morphisms in E(HR ) are YD H-module algebra isomorphisms. We are going to define a product in the category E(HR ). Let #R be the braided H product in the category MH R to differ from the braided product in Q . This makes sense when a YD H-module algebra A can be treated as an algebra in MH R forgetting the H-module structure of A and endowing with the induced H-module structure (1–5). Given two objects X and Y in E(HR ), we define a generalized cotensor product ∗ -bicomodules) as a subset of X#R Y : X ∧ Y (in terms of HR ½ X xi #yi ∈ X#R Y | X xi ⊳− h#yi = X ¾ xi #h −⊲ yi , ∀h ∈ HR . THE BRAUER GROUP OF A HOPF ALGEBRA 463 In the foregoing formula we may change the actions ⊳− and −⊲ of HR into the actions of H which are easier to check. P P X ∧ Y = { xi #yi ∈ X#R Y | h(1) · xi #h(2) ⊲1 yi (4–6) P = h(1) ⊲2 xi #h(2) · yi , ∀h ∈ H}. The formula (4–6) allow us to define a left H-action on X ∧ Y : X X X h· (xi #yi ) = h(1) · xi #h(2) ⊲1 yi = h(1) ⊲2 xi #h(2) · yi (4–7) P whenever xi #yi ∈ X ∧ Y and h ∈ H. The left H-action is YD compatible with the right diagonal H-coaction, so we obtain: Proposition 4.2.2. If X, Y are two objects of E(HR ), then X ∧ Y with the H-action (4–7) and the H-coaction inherited from X#R Y is a YD H-module algebra. Moreover , X ∧ Y is an object of E(HR ). Let H ∗ be the convolution algebra of H. There is a canonical YD H-module structure on H ∗ such that H ∗ is a YD H-module algebra: P h·p= p(1) < p(2) , h >, H-action (4–8) P ∗ ∗ h ·p= h(2) pS −1 (h∗(1) ), H-coaction for h∗ , p ∈ H ∗ and h ∈ H. One may easily check that H ∗ with the YD H-module structure (4–8) is an object in E(HR ), denoted I. Moreover I is the unit object of E(H) with respect to the product ∧. It follows that the category E(HR ) is a monoidal category. Denote by E(HR ) the set of the isomorphism classes of objects in E(HR ). The fact that E(HR ) is a monoidal category implies that the set E(HR ) is a semigroup. In general, E(HR ) is not necessarily a group. However, it contains a subgroup of a nice type. Recall that a YD H-module algebra A is said to be quantum commutative (q.c.) if X ab = b(0) (b(1) · a) (4–9) for any a, b ∈ A. That is, A is a commutative algebra in QH . Let X be a q.c. object in E(HR ). Let X be the opposite algebra in MH R . That is, X = X as a right H op -comodule, but with the multiplication given by X x◦y = y(0) x(0) R(y(1) ⊗ x(1) ) where x, y ∈ X. Since the H-action on X does not define an H-module algebra structure on X, we have to define a new H-action on X such that X together with the inherited H op -comodule structure is a YD H-module algebra. Let H act on X as follows: X h⇀x= hu(3) · (h(2) ⊲2 (h(5) ⊲1 x))R(S(h(4) ) ⊗ h(1) ) (4–10) 464 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG P P where h ∈ HR , x ∈ X, hu = S(h(2) )u−1 (h(1) ) and u = S(R(2) )R(1) ∈ H ∗ ∗ is the Drinfel’d element of H . Proposition 4.2.3. Let X be an object in E(HR ) such that X is q.c. Then: (1) X together with the H-action (4–10) is a YD H-module algebra. (2) X is a q.c. object in E(HR ) and X ∧ X ∼ = I = X ∧ X. Since the proof is lengthy, we refer reader to [59] for the complete proof. Denote by Gal(HR ) the subset of E(HR ) consisting of the isomorphism classes of objects in E(HR ) such that the objects are quantum commutative in QH . We have Theorem 4.2.4 [59, Thm. 3.12]. The set Gal(HR ) is a group with product induced by ∧ and inverse operator induced by H-opposite . 4.3. The exact sequence. For convenience we will call an H-Azumaya algebra A an R-Azumaya algebra if the H-action on A is of form (1–5). That is, A represents an element of BC(k, H, R). In this subsection we investigate the RAzumaya algebras which are Galois extensions of the coinvariants, and establish a group homomorphism from BC(k, H, R) to the group Gal(HR ) constructed in the previous subsection. In the sequel, we will write: X M0 = {m ∈ M | m(0) ⊗ m(1) = m ⊗ 1} for the coinvariant k-submodule of a right H-comodule M , in order to make a ∗ difference between HR -coinvariants and H-coinvariants. We start with a special elementary R-Azumaya algebra. Lemma 4.3.1. Let M = H op be the right H op -comodule, and let A be the elementary R-Azumaya algebra End(M ). Then A ∼ = H ∗ op #H op , where the left P H op -action on H ∗ op is given by h · p = p(1) hp(2) , S −1 (h)i = S −1 (h) ⇀ p, op ∗ op whenever h ∈ H and p ∈ H . Let A be an R-Azumaya algebra. We have [A#End(H op )] = [A] since End(H op ) represents the unit of BC(k, H, R). Now the composite algebra map λ H op −→ End(H op ) ֒→ A#End(H op ) is H op -colinear. It follows that A#End(H op ) is a smash product algebra B#H op where B = (A#End(H op ))0 . Thus we obtain that any element of BC(k, H, R) can be represented by an R-Azumaya algebra which is a smash product. Since any smash product algebra is a Galois extension of its coinvariants, we have that any element of BC(k, H, R) can be represented by an R-Azumaya algebra which is an H op -Galois extension of its coinvariants. Moreover, one may easily prove that if A is an R-Azumaya algebra such that it is an H op -Galois extension of A0 , then A is a H op -Galois extension of (A0 )op . THE BRAUER GROUP OF A HOPF ALGEBRA 465 An R-Azumaya algebra A is said to be Galois if it is a right H op -Galois extension of its coinvariant subalgebra A0 . Let A be a Galois R-Azumaya algebra. Denote by π(A) the centraliser subalgebra CA (A0 ) of A0 in A. It is clear that π(A0 ) is an H op -comodule subalgebra of A. The Miyashita–Ulbrich–Van Oystaeyen (MUVO) action (see [34; 48; 50; 51]; the last author mentioned considered it first in the situation of purely inseparable splitting rings in [50]) of H on π(A) is given by X h⇀a= Xih aYih (4–11) P h where Xi ⊗ Yih = β −1 (1 ⊗ h), for h ∈ H and β is the canonical Galois map P given by β(a ⊗ b) = ab(0) ⊗ b(1) . It is well-known (e.g., see [48; 11]) that π(A) together with the MUVO action (4–11) is a new YD H-module algebra. Moreover, π(A) is quantum commutative in the sense of (4–9) [48; 52]. Recall that when a Galois H op -comodule algebra A is an Azumaya algebra, the centraliser π(A) is a right H ∗ -Galois extension of k with respect to the MUVO action (4–11); compare [48]. This is not the case when A is an R-Azumaya ∗ -Galois object, instead of an algebra. However, π(A) turns out to be an HR ∗ H -Galois object. Proposition 4.3.2 [59, Prop. 4.5]. Let A be a Galois R-Azumaya algebra. Then ∗ π(A)/k is an HR -biextension and π(A) is an object in Gal(HR ). It is natural to expect the functor π to be a monoidal functor from the monoidal category of Galois R-Azumaya algebras to the monoidal category E(HR ). This is indeed the case. Proposition 4.3.3. π is a monoidal functor . That is: (1) If A and B are two Galois R-Azumaya algebras, then π(A#B) = π(A) ∧ π(B); and (2) If M is a finite right H op -comodule, and A = End(M ) is the elementary RAzumaya algebra such that A is a Galois R-Azumaya algebra, then π(A) ∼ = I. It follows that π induces a group homomorphism π e from the Brauer group BC(k, H, R) to the group Gal(HR ) sending element [A] to element [π(A)], where A is chosen as a Galois R-Azumaya algebra. In order to describe the kernel of π e, one has to analyze the H-coactions on the Galois R-Azumaya algebras. We obtain that the kernel of π e is isomorphic to the usual Brauer group Br(k). Thus we obtain the following exact sequence: Theorem 4.3.4 [59, Thm. 4.11]. We have an exact sequence of group homomorphisms: 1 ✲ Br(k) ι π e ✲ BC(k, H, R) ✲ Gal(HR ). Note that the exact sequence (4–12) indicates that the factor group BC(k, H, R)/Br(k) (4–12) 466 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG ∗ is completely determined by the HR -bigalois objects. In particular, when k is an algebraically closed field BC(k, H, R) is a subgroup of Gal(HR ). Now let us look at some special cases. First let H be a commutative Hopf algebra. H has a trivial coquasitriangular structure R = ε ⊗ ε. In this case. HR is equal to H as an algebra and D[H] = H ⊗H is the tensor product algebra. An R-Azumaya algebra is an Azumaya algebra which is a right H-comodule algebra with the trivial left H-action. On the other hand, the HR -bimodule structures (4–1) and (4–3) of a YD H-module M coincide and are exactly the left H-module structure of M . So in this case an object in the category E(HR ) is nothing but an H ∗ -Galois object which is automatically an H ∗ -bigalois object since H ∗ is cocommutative. So the group Gal(HR ) is the group E(H ∗ ) of H ∗ -Galois objects with the cotensor product over H ∗ . So we obtain the following exact sequence due to Beattie. Corrollary 4.3.5 [3]. Let H be a finite commutative Hopf algebra. Then the following group sequence is exact and split: 1 ✲ Br(k) ι π e ✲ BC(k, H) ✲ E(H ∗ ) −→ 1 where the group map π e is surjective and split because any H ∗ -Galois object B is equal to π(B#H) and the smash product B#H is a right H-comodule Azumaya algebra which represents an element in BC(k, H). Secondly we let R be a non-trivial coquasitriangular structure of H, but let H be a commutative and cocommutative finite Hopf algebra over k. In this case, HR is isomorphic to H as an algebra and becomes a Hopf algebra. In this case, an object in Gal(HR ) is an H ∗ -bigalois object. It is not difficult to check that YD H-module (or H-dimodule) structures commute with both H ∗ -Galois structures. Let θ be the Hopf algebra homomorphism corresponding to the coquasitriangular structure R, that is, θ : H −→ H ∗ , θ(h)(l) = R(l ⊗ h) for h, l ∈ H. Let ⇀ be the induced H-action on a right H-comodule M : X X h⇀m= m(0) θ(h)(m(1) ) = m(0) R(m(1) ⊗ h) for h ∈ H and m ∈ M . In [49], Ulbrich constructed a group D(θ, H ∗ ) consisting of isomorphism classes of H ∗ -bigalois objects which are also H-dimodule algebras such that all H and H ∗ structures commute, and satisfy the following additional conditions interpreted by means of R [49, (14), (16)]: P h −⊲ a = a(0) ⊳− h(1) R(a(1) ⊗ S(h(2) ))R(S(h(3) ) ⊗ a(2) ) P P x(0) (a ⊳− x(1) ) = (x(1) ⇀ a)x(0) , (4–13) THE BRAUER GROUP OF A HOPF ALGEBRA 467 Let us check that any object A in the category E(HR ) satisfies the conditions (4–13) so that A represents an element of D(θ, H ∗ ). Indeed, since H is commutative and cocommutative, we have X h −⊲ a = (h(2) · a(0) )R(S −1 (h(4) ) ⊗ h(3) a(1) S −1 (h(1) )) X = (h(1) · a(0) )R(S(h(2) ) ⊗ a(1) ) X = (h(2) · a(0) )R(a(1) ⊗ h(1) )R(a(2) ⊗ S(h(3) ))R(S(h(4) ) ⊗ a(3) ) X = (a(0) ⊳− h(1) )R(a(1) ⊗ S(h(2) ))R(S(h(3) ) ⊗ a(2) ), and X x(0) (a ⊳− x(1) ) = = = X X X x(0) (x(1) · a(0) )R(a(1) ⊗ x(1) ) a(0) x(0) R(a(1) ⊗ x(1) ) (by q.c.) (x(1) ⇀ a)x(0) for any a, x ∈ A and h ∈ H. It follows that the group Gal(HR ) is contained in D(θ, H ∗ ). As a consequence, we obtain Ulbrich’s exact sequence [49, 1.10]: π θ 1 −→ Br(k) −→ BD(θ, H ∗ ) −→ D(θ, H ∗ ) for a commutative and cocommutative finite Hopf algebra with a Hopf algebra homomorphism θ from H to H ∗ . 4.4. An example. Let k be a field with characteristic different from two. Let H4 be the Sweedler four dimensional Hopf algebra over k. That is, H4 is generated by two elements g and h satisfying g 2 = 1, h2 = 0, gh + hg = 0. The comultiplication, the counit and the antipode are as follows: ∆(g) = g ⊗ g, ∆(h) = 1 ⊗ h + h ⊗ g, ε(g) = 1, ε(h) = 0, S(g) = g, S(h) = gh. There is a family of CQT structures Rt on H4 parameterized by t ∈ k as follows: Rt 1 g h gh 1 g h 1 1 0 1 −1 0 0 0 t 0 0 t gh 0 0 −t t 468 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG It is not hard to check that the Hopf algebra homomorphisms Θl and Θr induced by Rt are as follows: Θl : H4cop −→ H4∗ , Θr : H4op −→ H4∗ , Θl (g) =, 1 − g, Θr (g) = 1 − g, Θl (h) = t(h − gh), Θr (h) = t(h + gh). When t is non-zero, Θl and Θr are isomorphisms, so that H4 is a self-dual Hopf algebra. We have that HRt is a 4-dimensional algebra generated by two elements u and v such that u and v satisfy the relations: u2 = 1, uv − vu = 0, v 2 = t(1 − u), which is isomorphic to the commutative algebra k[y]/hy 4 − 2ty 2 i when t is not zero. The double algebra D[H4 ] with respect to Rt is generated by four elements, g1 , g2 , h1 and h2 such that gi2 = 1, h2i = 0, gi hj + hj gi = 0, g1 g2 = g2 g1 , h1 h2 + h2 h1 = t(1 − g1 g2 ). The comultiplication of D[H4 ] is easy because the Hopf subalgebras generated by gi , hi , i = 1, 2, are isomorphic to H4 . Thus the algebra embedding φ reads as follows: HRt −→ D[H4 ], φ(u) = g1 g2 , φ(v) = g1 (h2 − h1 ). Let us consider the triangular case (H4 , R), where R = R0 . In this case, an algebra A is an H4 -module algebra if and only if it a DS-algebra (see section 2), and A is R-Azumaya algebra if and only if A is a DS-Azumaya algebra. From Theorem 2.5 we know that the Brauer group BC(k, H4 , R) is isomorphic to (k, +) × BW(k). Let us work out the group Gal(HR ) and calculate the Brauer group BC(k, H4 , R) using the sequence (4–12). First of all we have the following structure theorem of bigalois objects in E(HR ) [59]. Theorem 4.4.1 [59, Thm. 5.7]. Let A be a bigalois object in E(HR ). Then A is either of type (A) or of type (B ): ¡ ¢ Type (A): A is a generalized quaternion algebra α,β k , α 6= 0, with the following YD H-module structures: g · u = −u, h · u = 0, ρ(u) = u ⊗ 1 − 2uv ⊗ gh, g · v = −v, h · v = 1, ρ(v) = v ⊗ g + 2β ⊗ h. THE BRAUER GROUP OF A HOPF ALGEBRA 469 √ √ Type (B): A is a commutative algebra k( α) ⊗ k( β) with the following YD H-module structures: g · u = u, g · v = −v, h · u = 0, h · v = 1, ρ(u) = u ⊗ 1 + 2uv ⊗ h, ρ(v) = v ⊗ g + 2β ⊗ h, √ √ where k( α) and k( β) are generated by elements u and v respectively, and u, v satisfy the relations: u2 = α, uv = vu and v 2 = β. As a consequence of Theorem 4.4.1, we have the group structure of the group Gal(HR ): Proposition 4.4.2 [59, 5.8–5.9]. The group Gal(HR ) is equal to k × (k • /k • 2 ) × Z 2 as a set. The multiplication rule on the set is given by (β, α, i)(β ′ , α′ , j) = (β + β ′ , (−1)ij αα′ , i + j). The foregoing multiplication rule of Gal(HR ) shows that Gal(HR ) is a direct product of (k, +) and the group k • /k • 2 >⊳ Z 2 which is isomorphic to the group Q2 (k) of graded quadratic extensions of k (see [57]). Notice¡ that ¢ an object of type (A) in Gal(HR ) is some generalized quaternion algebra α,β with the H4 -action and coaction given in Theorem 4.4.1, where k ¡ ¢ • is a Galois R-Azumaya algebra if we α ∈ k and β ∈ k. When β 6= 0, α,β k ¡ ¢ is forget the left H4 -module structure. Since the coinvariant subalgebra of α,β k ¡ α,β ¢ ¡ α,β ¢ ¡ α,0 ¢ trivial, we have π( k ) = k if β 6= 0. To get an object k in Gal(HR ), ¡ ¢ ¡ 1,−1 ¢ where α ∈ k • , we consider the Galois R-Azumaya algebra α,1 . Since k # k π is monoidal, we have π µ³ ´ ³ 1, −1 ´¶ α, 1 k # k = ³ α, 1 ´ k ∧ ³ 1, −1 ´ k = ³ α, 0 ´ k for any α ∈ k • . √ √ For an object k( α) ⊗ k( β) of type ¡(B) ¢in Gal(HR ), we choose a Galois (assured by the foregoing arguR-Azumaya algebra A such that π(A) = α,β k ments). Then it is easy to check that p √ √ π(A#k( 1)) = k( α) ⊗ k( β) for α ∈ k • and β ∈ k. Thus we have shown that the homomorphism π̃ is surjective and we have an exact sequence: π̃ 1 −→ Br(k) −→ BC(k, H4 , R) −→ Gal(HR ) −→ 1. (4–14) Recall that the Brauer–Wall group BW(k) is BC(k, kZ 2 , R′ ), where kZ 2 is the sub-Hopf algebra of H4 generated by the group-like element g ∈ H4 , and R′ is the restriction of R to kZ 2 . The following well-known exact sequence is a special case of (4–12): π̃ 1 −→ Br(k) −→ BW(k) −→ Q2 (k) −→ 1, (4–15) 470 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG where Q2 (k) is nothing but Gal(HR′ ) and HR′ ∼ = kZ 2 , here H = kZ 2 . The sequence (4–15) can be also obtained if we restrict the homomorphism π̃ in (4–14) to the subgroup BW(k) of BC(k, π̃(BW(k)) √ of ¡ H4¢, R). The subgroup √ Gal(HR ) consists of all objects of forms: α,0 of type (A) and k( α) ⊗ k( 0) k of type (B), which is isomorphic to Q2 (k) (see [59] for details). In fact we have the following commutative diagram: 1 ✲ Br(k) ∩ K ✲ K π̃ ✲ (k, +) ✲ 1 ι 1 ❄ ✲ Br(k) ❄ ❄ π̃ ✲ BC(k, H4 , R) ✲ Gal(HR0 ) ✲ 1 γ 1 ❄ ✲ Br(k) ❄ ✲ BW(k) p π̃ ✲ ❄ Q2 (k) ✲ 1, where γ is the canonical map defined in Theorem 2.5, K is the kernel of γ, ι is the inclusion map and p is the projection from (k, +) × Q2 (k) onto Q2 (k). Here π̃(K) = (k, +) because¡ π̃ ◦¢γ = p¡ ◦ π̃¢ (which √ can be easily checked on α,β 1)). By definition of γ we Galois R-Azumaya algebras α,β and #k( k k (k, +). Since γ is split, we obtain have Br(k) ∩ K = 1. It follows that K ∼ = that the Brauer group BC(k, H4 , R) is isomorphic to the direct product group (k, +) × BW(k), which coincides with Theorem 2.5. Recently, G. Carnovale proved in [13] that the Brauer group BC(k, H4 , Rt ) is isomorphic to BC(k, H4 , R0 ) for any t 6= 0 although (H4 , Rt ) is not coquasitriangularly isomorphic to (H4 , R0 ) when t 6= 0 [40]. 5. The Hopf Automorphism Group Let H be a faithfully projective Hopf algebra over a commutative ring k. As we have seen from the previous section, the Brauer group BQ(k, H) may be approximated by computing the group Gal(HR ), where HR is a deformation of the dual D(H)∗ of the quantum double D(H). However, to compute explicitly the group BQ(k, H) is a hard task. On the other hand, there are some subgroups of BQ(k, H) which are (relatively) easier to calculate. For instance, when H is commutative and cocommutative, various subgroups of the Brauer–Long group could more easily be studied [3; 4; 7; 8; 10; 17]. One of these subgroups is Deegan’s subgroup introduced in [17] which involves the Hopf algebra structure of H itself and in fact turns out to be isomorphic to the Hopf algebra automorphism group Aut(H) [17; 8]. The connection between Aut(H) and BD(k, H) for some particular commutative and cocommutative Hopf algebra H was probably THE BRAUER GROUP OF A HOPF ALGEBRA 471 first studied by M.Beattie in [3] where she established the existence of an exact sequence: β 1 −→ BC(k, G)/Br(k) × BM(k, G)/Br(k) −→ B(k, G)/Br(k) −→ Aut(G) −→ 1 where B(k, G) is the subgroup of BD(k, G) consisting of the classes represented by G-dimodule Azumaya algebras whose underlying algebras are Azumaya, Aut(G) is the automorphism group of G, G is a finite abelian group and k is a connected ring. Based on Beattie’s construction of the map β, Deegan constructed his subgroup BT(k, G) which is then isomorphic to Aut(G); the resulting embedding of Aut(G) in the Brauer–Long group (group case) is known as Deegan’s embedding theorem. In [8], S.Caenepeel looked at the Picard group of a Hopf algebra, and extended Deegan’s embedding theorem from abelian groups to commutative and cocommutative Hopf algebras. But if H is a quantum group (i.e., a (co)quasitriangular Hopf algebra) or just any non-commutative non-cocommutative Hopf algebra then it seems that the method of Deegan and Caenepeel cannot be extended to obtain a group homomorphism from some subgroup of BQ(k, H) to the automorphism group Aut(H). In fact, Aut(H) can no longer be embedded into BQ(k, H). On the other hand, the idea of Deegan’s construction can still be applied to our non-commutative and non-cocommutative case. Let M be a faithfully projective Yetter–Drinfel’d H-module. Then Endk (M ) is an H-Azumaya YD H-module algebra. However, if M is an H-bimodule, that is, a left H-module and a right H-comodule, but not a YD H-module, it may still happen that Endk (M ) is a YD H-module algebra. Take a non-trivial Hopf algebra isomorphism α ∈ Aut(H) (for example, if the antipode S of H is not of order two, S 2 is a non-trivial Hopf automorphism). We define a left H-module and a right H-comodule Hα as follows: as a k-module Hα = H; we equip Hα with the obvious H-comodule structure given by ∆, and an H-module structure given by X h·x= α(h(2) )xS −1 (h(1) ) for h ∈ H, x ∈ Hα . Since α is nontrivial Hα is not a YD H-module. Let Aα = End(Hα ) with H-structures induced by the H-structures of Hα , that is, X (h · f )(x) = h(1) f (S(h(2) ) · x) X χ(f )(x) = f (x(0) )(0) ⊗ S −1 (x(1) )f (x(0) )(1) for f ∈ Aα , x ∈ Hα . Lemma 5.1 [12, 4.6, 4.7]. If H is a faithfully projective Hopf algebra and α is a Hopf algebra automorphism of H, then Aα is an Azumaya YD-module algebra and the following map defines a group homomorphism: ω : Aut(H) −→ BQ(k, H), α 7→ [Aα−1 ]. 472 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG In the sequel, we will compute the kernel of the map ω. Let D(H) denote the Drinfel’d double of H. Let A be an H-module algebra. Recall from [5] that the H-action on A is said to be strongly inner if there is an algebra map f : H −→ A such that X h·a= f (h(1) )af (S(h(2) )), a ∈ A, h ∈ H. Lemma 5.2. Let M be a faithfully projective k-module. Suppose that End(M ) is a D(H)-Azumaya algebra. Then [End(M )] = 1 in BM(k, D(H)) if and only if the D(H)-action on A is strongly inner . The proof has its own interest. Suppose that the D(H)-action on A is strongly inP ner. There is an algebra map f : D(H) → A such that t·a = f (t(1) )af (S(t(2) )), t ∈ D(H), a ∈ A. This inner action yields a D(H)-module structure on M given by t ⇀ m = f (t)(m), t ∈ D(H), m ∈ M. Since f is an algebra representation map the above action does define a module structure. Now it is straightforward to check that the D(H)-module structure on A is exactly induced by the D(H)-module structure on M defined above. By definition [End(M )] = 1 in BM(k, D(H)). Conversely, if [A] = 1, then there exists a faithfully projective D(H)-module N such that A ∼ = End(N ) as D(H)-module algebras by [12, 2.11]. Now D(H) acts strongly innerly on End(N ). Let u : D(H) −→ End(N ) be the algebra representation map. Now one may easily verify that the strongly inner action induced by the composite algebra map: u µ : D(H) −→ End(N ) ∼ =A exactly defines the D(H)-module structure on A. Lemma 5.3. For a faithfully projective k-module M , let u, v : H −→ End(M ) define H-module structures on M , call them Mu and Mv . If End(Mu ) = End(Mv ) as left H-modules via (1–2), then (v ◦ S) ∗ u is an algebra map from H to k, i .e., a grouplike element in H ∗ . Similarly, if M admits two H-comodule structures ρ, χ such that the induced H-comodule structures on End(M ) given by (1–2) coincide, then there is a grouplike element g ∈ G(H) such that χ = (1 ⊗ g)ρ, i .e., P P χ(x) = x(0) ⊗ gx(1) if ρ(x) = x(0) ⊗ x(1) for x ∈ M . Proof. For any m ∈ M, h ∈ H, φ ∈ End(Mu ) = End(Mv ), X X u(h(1) )[φ[u(S(h(2) ))(m)]] = v(h(1) )[φ[v(S(h(2) ))(m)]], or equivalently, X v(S(h(1) ))[u(h(2) )(φ[u(S(h(3) ))(m)])] = φ(v(S(h))(m)). Let λ = (v ◦ S) ∗ u : H −→ End(M ) with convolution inverse (u ◦ S) ∗ v. Letting m = u(h(4) )(x) for any x ∈ H in the equation above, we obtain λ(h) ∈ THE BRAUER GROUP OF A HOPF ALGEBRA 473 Z(End(M )) = k for all h ∈ H. Since u, v are algebra maps, it is easy to see that λ is an algebra map from H to k. ¤ Given a group-like element g ∈ G(H), g induces an inner Hopf automorphism of H denoted g, i.e., g(h) = g −1 hg, h ∈ H. Similarly, if λ is a group-like element of H ∗ , then λ induces a Hopf automorphism of H, denoted by λ where λ(h) = P λ(h(1) )h(2) λ−1 (h(3) ), h ∈ H. Since G(D(H)) = G(H ∗ ) × G(H) ([40, Prop.9]) and g commutes with λ in Aut(H), we have a homomorphism θ: G(D(H)) −→ Aut(H), (λ, g) 7→ gλ. Let K(H) denote the subgroup of G(D(H)) consisting of elements {(λ, g) | g −1 (h) = λ(h), ∀h ∈ H}. Lemma 5.4. Let H be a faithfully projective Hopf algebra. Then K(H) ∼ = G(D(H)∗ ). Proof. By [40, Prop.10], an element g ⊗ λ is in G(D(H)∗ ) if and only if g ∈ G(H), λ ∈ G(H ∗ ) and g, λ satisfy the identity: g(λ ⇀ h) = (h ↼ λ)g, ∀h ∈ H, P where, λ ⇀ h = h(1) λ(h(2) ) and h ↼ λ = h(2) λ(h(1) ). Let g ∈ G(H), λ ∈ ∗ G(H ), for any h ∈ H, we have X X X X gh(1) λ(h(2) ) = λ(h(1) )h(2) g ⇐⇒ h(1) λ(h(2) ) = λ(h(1) )g −1 h(2) g X X ⇐⇒ λ−1 (h(1) )h(2) λ(h(3) ) = g −1 hg. P This means g ⊗ λ is in G(D(H)∗ ) if and only if (λ, g) ∈ K(H). Therefore K(H) = G(D(H)∗ ). ¤ Applying Lemmas 5.1–5.4, one may able to show that the group homomorphism θ can be embedded into the following long exact sequence: Theorem 5.5 [52, Thm. 5]. Let H be a faithfully projective Hopf algebra over k. The following sequence is exact: θ ω 1 −→ G(D(H)∗ ) −→ G(D(H)) −→ Aut(H) −→ BQ(k, H), (5–1) where θ(λ, g) = λg and ω(α) = Aα−1 = End(Hα−1 ). As a consequence of the theorem, we rediscover the Deegan–Caenepeel’s embedding theorem for a commutative and cocommutative Hopf algebra [8; 17]. Corrollary 5.6. Let H be a faithfully projective Hopf algebra such that G(H) and G(H ∗ ) are contained in the centers of H and H ∗ respectively. Then the map ω in the sequence (5–1) is a monomorphism. In particular , if H is a commutative and cocommutative faithfully projective Hopf algebra over k, then Aut(H) can be embedded into BQ(k, H). 474 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG Note that in this case, G(D(H)∗ ) = G(D(H)). It follows that the homomorphism θ is trivial , and hence the homomorphism ω is a monomorphism. In the following, we present two examples of the exact sequence (5–1). Example 5.7. Let H be the Sweedler Hopf algebra over a field k in Subsection 4.4. H is a self-dual Hopf algebra, i.e., H ∼ = H ∗ as Hopf algebras. It is straightforward to show that the Hopf automorphism group Aut(H) is isomorphic to k • = k\0 via: f ∈ Aut(H), f (g) = g, f (h) = zh, z ∈ k • . Considering the group G(D(H)) of group-like elements, it is easy to see that G(D(H)) = {(ε, 1), (λ, 1), (ε, g), (λ, g)} ∼ = Z2 × Z2 where λ = p1 − pg , and p1 , pg is the dual basis of 1, g. One may calculate that the kernel of the map θ is given by: K(H) = {(ε, 1), (λ, g)} ∼ = Z2 The image of θ is {1, g} which corresponds to the subgroup {1, −1} of k ∗ . Thus by Theorem 5.5 we have an exact sequence: 1 −→ Z 2 −→ Z 2 × Z 2 −→ k • −→ BQ(k, H), It follows that k • /Z 2 can be embedded into the Brauer group BQ(H). In particular, if k = R, the real field, then Br(R) = Z 2 ⊂ BQ(R, H), and R• /Z 2 is a non-torsion subgroup of BQ(R, H). In the previous example, the subgroup k • /Z 2 of the Brauer group BQ(k, H) is still an abelian group. The next example shows that the general linear group GLn (k) modulo a finite group of roots of unity for any positive number n may be embedded into the Brauer group BQ(k, H) of some finite dimensional Hopf algebra H. Example 5.8. Let m > 2, n be any positive numbers. Let H be Radford’s Hopf algebra of dimension m2n+1 over C (complex field) generated by g, xi , 1 ≤ i ≤ n such that g 2m = 1, xi 2 = 0, gxi = −xi g, xi xj = −xj xi . The coalgebra structure ∆ and the counit ε are given by ∆g = g ⊗ g, ∆xi = xi ⊗ g + 1 ⊗ xi , ε(g) = 1, ε(xi ) = 0, 1 ≤ i ≤ n. By [40, Prop.11], the Hopf automorphism group of H is GLn (C). Now we compute the groups G(D(H)) and G(D(H)∗ ). It is easy to see that G(H) = (g) (see also [40, p353]) is a cyclic group of order 2m. Let ωi , 1 ≤ i ≤ m be the m-th roots of 1, and let ζj be the m-th roots of −1. Define the algebra maps ηi and λi from H to C as follows: ηi (g) = ωi g, ηi (xj ) = 0, 1 ≤ i, j ≤ m, THE BRAUER GROUP OF A HOPF ALGEBRA 475 and λi (g) = ζi g, λi (xj ) = 0, 1 ≤ i, j ≤ m. One may check that {ηi , λi }ni=1 is the group G(H ∗ ). It follows that G(D(H)) = G(H) × G(H ∗ ) ∼ = (g) × U , where U is the group of 2m-th roots of 1. To compute G(D(H)∗ ) it is enough to calculate K(H). Since gi where ν(g) = g, = ½ id if i is even, ν if i is odd, ν(xj ) = −xj , 1 ≤ j ≤ n, and ηi (g) = g, ηi (xj ) = ωi xj , λi (g) = g, λi (xj ) = ζi xj , 1 ≤ i, j ≤ n, 1 ≤ i, j ≤ n. It follows that K(H) = {(ε, g 2i ), (ψ, g 2i−1 ), 1 ≤ i ≤ m}, where ψ is is given by: ψ(g) = −g, ψ(xi ) = 0. Consequently G(D(H)∗ ) ∼ = U , Since the base field is C, (g) ∼ = U , and we have an exact sequence 1 −→ U −→ U × U −→ GLn (C) −→ BQ(C, H). The two examples above highlight the interest of the study of the Brauer group of a Hopf algebra. In Example 5.8, even though the classical Brauer group Br(C) is trivial, the Brauer group BQ(C, H) is still large enough. In the rest of this section, we consider a natural action of Aut(H) on BQ(k, H). Let A be an H-Azumaya algebra, and α a Hopf algebra automorphism of H. Consider the YD H-module algebra A(α), which equals A as a k-algebra, but with H-structures of (A(α), ⇁, χ′ ) given by X h ⇁ a = α(h) · a and χ′ (a) = a(0) ⊗ α−1 (a(1) ) = (1 ⊗ α−1 )χ(a) for all a ∈ A(α), h ∈ H. Lemma 5.9. Let A, B be H-Azumaya algebras. If α is a Hopf algebra automorphism of H, then A(α) is an H-Azumaya algebra, and (A#B)(α) ∼ = A(α)#B(α). The action of Aut(H) on BQ(k, H) is an inner action. Indeed, we have: Theorem 5.10 [12, Thm. 4.11]. Aut(H) acts innerly on BQ(k, H), more precisely, for any H-Azumaya algebra B and α ∈ Aut(H), we have [B(α)] = [Aα ][B][Aα−1 ]. 476 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG Theorem 5.10 yields the multiplication rule for two elements [B] and ω(α) = [Aα ] where B is any H-Azumaya algebra and α is in Aut(H). In particular, if T is a subgroup such that T is invariant (or stable) under the action of Aut(H), then the subgroup generated by T and ω(Aut(H)) is (ι ⊗ ω)(T >⊳ Aut(H)), where >⊳ is the usual semi-direct product of groups. Example 5.11. Let (H4 , R0 ) be the CQT Hopf algebra described in Subsection 4.4. The automorphism group Aut(H4 ) of H4 is isomorphic to k • . If A is an R0 Azumaya algebra and α ∈ Aut(H4 ), then A(α) is still an R0 -Azumaya algebra as the automorphism α does not affect the induced action (1–5). Thus the subgroup BC(k, H4 , R0 ) is stable under the action of Aut(H4 ). By Example 5.7 we get a non-abelian subgroup of BQ(k, H4 ) (see [56]): (ι ⊗ ω)(BC(k, H4 , R0 ) >⊳ k • ) ∼ = BC(k, H4 , R0 ) >⊳ (k • /Z 2 ) ∼ = BW(k) × (k, +) >⊳ (k • /Z 2 ). 6. The Second Brauer Group In the classical Brauer group theory of a commutative ring k, an Azumaya algebra can be characterized as a central separable algebra over k. However this is not the case when we deal with the Brauer group of structured algebras. For instance, in the Brauer–Wall group of a commutative ring k, a representative (i.e., a Z 2 -graded Azumaya algebra) is not necessarily a central separable algebra, instead it is a graded central and graded separable algebra. Motivated by the example of the Brauer–Wall group, one may be inspired to try to define the Brauer group of a Hopf algebra H by using H-separable algebras in a natural way as we did for the Brauer-long group of Z 2 -dimodule algebras in section 2. In 1974, B. Pareigis for the first time defined two Brauer groups in a symmetrical category (see [38]). The first Brauer group was defined in terms of Morita equivalence whereas the second Brauer group was defined by so called ‘central separable algebras’ in the category. The two Brauer groups happen to be equal if the unit of the symmetrical category is a projective object. This is the case if the category is the Z 2 -graded module category with the graded product (2–2). However the dimodule category of a finite abelian group (or a finite commutative cocommutative Hopf algebra) is not a symmetric category, instead a braided monoidal category ([32]). Therefore, the definition of the second Brauer group due to Pareigis can not be applied to the dimodule category. Nevertheless, we are able to modify Pareigis’s definitions to get the proper definitions of the two Brauer groups for a braided monoidal category (see [52]) so that they allow to recover all known Brauer groups. We are not going into the details of the categorical definitions given in [52]. Instead we will focus our attention on the Yetter Drinfel’d module category of a Hopf algebra. Like the Brauer–Wall group BW(k), the Brauer–Long group BD(k, H) of a finite commutative and cocommutative Hopf algebra H over a commutative ring THE BRAUER GROUP OF A HOPF ALGEBRA 477 k can be defined by central separable algebras in the category when k is nice (e.g., k is a field with characteristic 0. However, a counter example exists when k is not so nice, e.g., BD(k, Z 2 ) when 2 is not a unit in k (see [7]). When a Hopf algebra is not commutative and cocommutaive, even if k is a field with ch(k) = 0, the second Brauer group defined by H-separable algebras turns out to be smaller than the Brauer group of H-Azumaya algebras. In other words, an H-Azumaya algebra is not necessarily an H-separable algebra. Some examples of this will be presented. Let H be a Hopf algebra over k and let Ae (or e A) be the H-enveloping algebra A#A (or A#A). Definition 6.1. Let A be a YD H-module algebra. A is said to be H-separable if the following exact sequence splits in Ae QH : π̃ A Ae −→ A −→ 0. In this section πA is the usual multiplication of A. We will often use M0 to T stand for M H M coH , the intersection of the invariants and the coinvariants of YD H-module M . A H-separable algebra can be described by separability idempotent elements. Proposition 6.2. Let A be a YD H-module algebra. The following statements are equivalent: (1) A is H-separable. (2) There exists an element el ∈ Ae0 such that πA (el ) = 1 and (a#1)el = (1#a)el for all a ∈ A. (3) There exists an element e ∈ (A#A)0 such that πA (el ) = 1 and (a#1)e = e(1#a) for all a ∈ A. (4) There exists an element er ∈ e A0 such that πA (er ) = 1 and er (a#1) = er (1#a) for all a ∈ A. (5) πA : e A −→ A −→ 0 splits in QHe A . The proof of these statements is straightforward. We emphasize that H-separable algebras are k-separable by the statement (3). However a separable YD Hmodule ­algebra ® is not necessarily H-separable. For instance, the H4 -Azumaya 1,−1 is a separable algebra and kZ 2 -separable, but not a H4 -separable algebra k algebra. We also have that H4 itself is an H4 -Azumaya algebra, but it is certainly not a separable algebra over any field. P If el = xi #y i is a separability idempotent in A#A, we may choose e = P P xi #yi and er = xi #yi . Thus we may write eA for el and e′A for er without ambiguity. Since eA is an idempotent element in each of the above cases, it follows that if A is H-separable then M A = eA ⇀ M and A N = N ↼ e′A for M ∈ Ae QH and N ∈ QHe A respectively. In particular, AA = eA ⇀ A and A A = A ↼ e′A . For a YD H-module algebra A we shall call AA and A A the left and the right H-center of A respectively. In case AA = k or A A = k we shall 478 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG say that A is left or right central respectively, and A is H-central if A is both left and right central. Proposition 6.3. (1) Let f : A −→ B be an epimorphism of YD H-module algebras. If A is H-separable then B is H-separable. (2) Let E be a commutative k-algebra. If A is H-separable then E ⊗k A is an E ⊗k H-separable E-algebra. (3) If A is H-separable, then A is H-separable. If in addition, A is left (or right) central then A is right (or left) central respectively. (4) If A, B are H-separable, so is A#B. If in addition, A and B are left (or right) central, then A#B is left (or right) central. Like the classical case, the ground ring k is an H-direct summand of of a left (or right) central H-separable algebra. Lemma 6.4. Let A be a left or right H-central H-separable algebra. Then the inclusion map embeds k as a direct summand of A in QH . Proof. Let e be an H-separability idempotent of A. Then the map Te : A −→ k given by Te (a) = e ⇀ a for a ∈ A is a YD H-module map. We have e ⇀ 1 = πA (e) = 1. ¤ The map Te described above is a section for the inclusion map ι : k ֒→ A in QH , that is, Te ◦ ι = id. We will call a YD H-module map T : A −→ k an H-trace map of a YD H-module algebra A if T (1) = 1. Notice that usually a trace map is an onto map but does not necessarily carry the unit to the unit. We will show later that an H-Azumaya algebra A is an H-central H-separable algebra if and only if A has an H-trace map. It follows that H-trace maps in a one-to-one way correspond to H-separability idempotents when A is an H-Azumaya algebra. A YD H-module algebra A is said to be H-simple if A has no proper YD H-module ideal (simply H-ideal). This is equivalent to A being simple in Ae QH or QHe A . Proposition 6.5. Let A be a left (or right) H-central H-separable algebra. Then A is H-simple if and only if k is a field . Proof. Suppose that A is H-simple and I is a non-zero ideal of k. IA is an H-ideal of A and IA = A. Let t be the H-trace map described in Lemma 6.4. Then t(IA) = t(A) implies I = k. It follows that k is a field. Conversely, suppose that A is an H-separable algebra over a field k. Since H-separability implies k-separability, A is semisimple artinian. Let M be an Hideal of A, then there exists a central idempotent c ∈ A such that M = cA = Ac. c must be in A0 . Now for any a ∈ A, we have X X a(0) (a(1) · c) = ac = ca, c(0) (c(1) · a) = ca = ac. Thus c is in both AA and A A. Now if A is left or right H-central H-separable algebra then c ∈ k, and hence M = cA = A. ¤ THE BRAUER GROUP OF A HOPF ALGEBRA 479 Lemma 6.6. If A is a left or right H-central H-separable algebra, then for any maximal H-ideal I of A there exists a maximal ideal α of k such that I = αA and I ∩ k = α. In view of Proposition 6.5 and Lemma 6.6, one may use an arqument similar to the classical case [14, 2.8] to obtain that a H-central H-separable algebra is an H-Azumaya algebra with an H-trace map. In fact, we have the following: Theorem 6.7. A YD H-module algebra A is an H-central H-separable algebra if and only if A is an H-Azumaya algebra with an H-trace map. Proof. By the foregoing remark, it is sufficient to show that an H-Azumaya algebra with an H-trace map is H-central and H-separable. Assume that A is an H-Azumaya algebra with an H-trace map T . Since A is H-Azumaya we have the isomorphism A#A ∼ = End(A). In this way we may view T as an element in A#A. In fact T is in (A#A)0 since T is H-linear and H-colinear. We now can show that T is an H-separability idempotent of Ae . Now πA (T ) = T (1) = 1, and for any a, x ∈ A, (a#1)T (x) = aT (x) = (1#a)T (x), because T (x) ∈ k. It follows from the foregoing equalities that we have (a#1)T = (1#a)T for any a ∈ A. Therefore, A is H-separable. ¤ In general, an H-Azumaya algebra is not necessarily H-separable, in other words, an H-Azumaya algebra need not have an H-trace map. For example, H4 is not a separable algebra, but it is an H4 -Azumaya algebra (see [56]). For this reason, we call an H-central H-separable algebra a strongly H-Azumaya algebra (for short we say that it is strong). Corrollary 6.8. Let A, B be H-Azumaya algebras. If A#B is strong, so are A and B. Proof. By Theorem 6.7 it is enough to show that both A and B have an Htrace map. This is the case since A#B has an H-trace map T and the restriction map TA (a) = T (a#1) and TB are clearly H-trace maps of A and B respectively. ¤ This corollary indicates that even the trivial H-Azumaya algebra End(M ), for M a faithfully projective YD H-module, is not necessarily strong. For example, if A is non-strongly H-Azumaya, e.g., A = H4 , then A is not strong, and hence End(A) ∼ = A#A is not strong by Corollary 6.8. So a strongly H-Azumaya algebra may be Brauer equivalent to a non-strongly H-Azumaya algebra. Now a natural question arises. What condition has to be imposed on H so that any H-Azumaya algebra is strongly H-Azumaya? We have a complete answer for a faithfully projective Hopf algebra, and a partial answer for an infinite Hopf algebra over a field with characteristic 0. 480 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG Proposition 6.9. Let H be a faithfully projective Hopf algebra over k. The following are equivalent: (1) Any H-Azumaya algebra is strongly H-Azumaya. (2) Any elementary H-Azumaya algebra is strongly H-Azumaya. (3) There exist an integral t ∈ H and an integral ϕ ∈ H ∗ such that ε(t) = 1 and ϕ(1) = 1. (4) k is a projective object in QH . Proof. (1) ⇐⇒ (2) due to Corollary 6.8. To prove that (2) =⇒ (3), we take the faithfully projective YD H-module M which is the left regular H-module of H itself, with the H-comodule structure given by X ρ(h) = h(2) ⊗ h(3) S −1 (h(1) ) for any h ∈ M . Now let A be the elementary H-Azumaya algebra End(M ). Since A is strong, A has an H-trace map, say, T : A −→ k. Since A is faithfully projective, A∗ is a YD H-module. We may view T as an element in A∗0 as T is ˜ ∗ M as a YD H-module. One may H-linear and H-colinear. Identify A∗ with M ⊗ ∗H easily check that the k-module A of H-invariants of A∗ consists of elements of form ¾ ½ Z X ∗ t(1) ⊗ t(2) ⇀ f | t ∈ , f ∈ M l R where l is the rank one k-module of left integrals of H and (t(2) ⇀ f )(h) = P f (S −1 (t(2) )h) for any h ∈ M . It follows that T = t(1) ⊗ t(2) ⇀ f for some left integral t ∈ H and an element f in ∗ M . Let {mi ⊗ pi } be a dual basis of M so P that mi ⊗ pi = 1A . Since T (1A ) = 1k , we obtain: X X T (1A ) = pi (t(1) )f (S −1 (t(2) )mi ) = f (S −1 (t(2) )t(1) ) = ε(t)f (1) = 1. So ε(t) is a unit of k, and one may choose a left integral t′ to replace t so that ε(t′ ) = 1. Similarly, if we choose a faithfully projective YD H-module M as follows: M = H as a right H-comodule with the comultiplication as the right comodule structure and with the adjoint left H-action given by X h·m= h(2) mS −1 (h(1) ), then one may find a left integral ϕ ∈ H ∗ such that ϕ(1) = 1. (3) ⇐⇒ (4) is the Maschke theorem. Since H is a faithfully projective Hopf algebra, the quantum double Hopf algebra D(H) is faithfully projective over k, and D(H) is a projective object in QH . Assume that there are two left integrals t ∈ H and ϕ ∈ H ∗ such that ε(t) = 1 and ϕ(1) = 1. The counit of D(H) is a YD H-module map which is split by the YD H-module map ι′ : k −→ D(H) sending the unit 1 to the element ϕ ⊲⊳ t. So k is a YD H-module direct summand of D(H), and hence it is projective in QH . The converse holds as the foregoing argument can be reversed. THE BRAUER GROUP OF A HOPF ALGEBRA 481 Finally, we show that (3) =⇒ (1). Assume ϕ ⊲⊳ t is a left integral of D(H) such that ε(t) = 1 = ϕ(1). If A is an H-Azumaya algebra, then the multiplication map π : A#A −→ A splits as a left A#A-module. Let µ : A −→ A#A be the split map, and let e = µ(1). Then (ϕ ⊲⊳ t) · e is an H-separability element of A. So A is strongly H-Azumaya. ¤ If H is not a faithfully projective Hopf algebra, we have a sufficient condition which requires that the antipode of H be involutory. Proposition 6.10. Let k be a field with ch(k) = 0 and let H be a Hopf algebra over k. If the antipode S of H is involutory, then any H-Azumaya algebra is strongly H-Azumaya. Proof. By Corollary 6.8, it is enough to show that any elementary H-Azumaya algebra is strong. Let M be a faithfully projective YD H-module, and let A = End(M ). We show that A has an H-trace map. Identify A with M ⊗ M ∗ as YD H-modules. Let tr be the normal trace map of A which sends the element m ⊗ m∗ to m∗ (m). Since ch(k) = 0, we have that tr(1A ) = n for some integer is a unit. We show that tr is a YD H-module map, then the statement follows. Indeed, if h ∈ H, m ∈ M and m∗ ∈ M ∗ , we have X X tr(h · (m ⊗ m∗ )) = tr(h(1) · m ⊗ h(2) · m∗ ) = (h(2) · m∗ )(h(1) · m) X = m∗ (S(h(2) )h(1) · m) = ε(h) tr(m). Similarly, tr is H-colinear as well. ¤ Note that when ch(k) = 0 the condition S 2 = id is equivalent to the condition (3) in Proposition 6.9 if H is finite dimensional (see [28]). However, this is not the case when H is not finite. There is an example of a Hopf algebra that is involutory (e.g., T (V ), the universal enveloping Hopf algebra), but without integrals. Nevertheless, it remains open whether k is a projective object in QH if and only if S 2 = id in case ch(k) = 0. As mentioned in the title of this section, we are able to define the second Brauer group of strongly H-Azumaya algebras as a result of proposition 6.3. That is, the second Brauer group, denoted BQs(k, H), consists of isomorphism classes of strongly H-Azumaya algebras modulo the same Brauer equivalence, where the elementary H-Azumaya algebras End(M ) are required to be H-separable. It is evident from Theorem 6.7 that the second Brauer group BQs(k, H) is a subgroup of BQ(k, H), which contains the usual Brauer group Br(k) as a normal subgroup. Let us summarize it as follows: Corrollary 6.11. The subset BQs(k, H) represented by the strongly H-Azumaya algebras is a subgroup of BQ(k, H). 482 FREDDY VAN OYSTAEYEN AND YINHUO ZHANG Proof. The only thing left to check is the coincidence of the two Brauer equivalence relations. Assume that A and B are two strongly H-Azumaya algebras and [A] = [B] in BQ(k, H). Then A#B ∼ = End(M ) for some faithfully projective YD H-module M . It follows that A#End(B) ∼ = B#End(M ). By Proposition 6.3, End(B) and End(M ) are strongly H-Azumaya algebras, and we obtain that [A] = [B] in BQs(k, H). ¤ Now the question arises: when is BQs(k, H) equal to BQ(k, H)? When a Hopf algebra satisfies the assumption in Proposition 6.9 or Proposition 6.10, BQ(k, H) = BQs(k, H). However, since a strongly H-Azumaya algebra may be Brauer equivalent to a non-strongly H-Azumaya algebra, there is a possibility that for some Hopf algebra H, any BQ(k, H) element can be represented by a strongly H-Azumaya algebra, but at the same time there may exist non-strongly H-Azumaya algebras. Note that for a quasitriangular or coquasitriangular Hopf algebra H, The H-central and H-separable or strongly H-Azumaya algebras are special cases of those above. For example, If (H, R) is a cosemisimple-like coquasitriangular Hopf algebra, then BCs(k, H, R) = BC(k, H, R). 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