On cotriangular Hopf algebras (original) (raw)

2001, American Journal of Mathematics

finite-dimensional right comodules are non-negative integers (maybe after modifying the Rform). This enables us to apply the same theorem of Deligne on Tannakian categories [De] that we applied in the proof of Theorem 2.1 from [EG1]. In Section 5, we give examples of twisted function algebras. In particular, we show that in the infinite-dimensional case, the squared antipode for such an algebra may not equal the identity (see Example 5.2 below). In Section 6, we show that in all of our examples, the operator S 2 is unipotent on A, and conjecture it to be the case for any twisted function algebra. We prove this conjecture, using the quantization theory of [EK1-2], in a large number of special cases. In Section 7, we formulate a few open questions. Throughout the paper, k will denote an algebraically closed field of characteristic 0. Acknowledgements The first author is grateful to Ben Gurion University for its warm hospitality, and to Miriam Cohen and the Dozor Fund for making his visit possible; his work was also supported by the NSF grant DMS-9700477. The second author is grateful to Susan Montgomery for numerous useful conversations. The authors would like to acknowledge that this paper was inspired by the work [BFM]. 2 Hopf 2−cocycles Let A be a coassociative coalgebra over k. For a ∈ A, we write ∆(a) = a 1 ⊗ a 2 , (I ⊗ ∆)∆(a) = a 1 ⊗ a 2 ⊗ a 3 etc, where I denotes the identity map of A. Recall that A * is an associative algebra with product defined by (f * g)(a) = f (a 1)g(a 2). This product is called the convolution product. Now let (A, m, 1, ∆, ε, S) be a Hopf algebra over k. Recall [Do] that a linear form J : A ⊗ A → k is called a Hopf 2−cocycle for A if it has an inverse J −1 under the convolution product * in Hom k (A ⊗ A, k), and satisfies: J(a 1 b 1 , c)J(a 2 , b 2) = J(a, b 1 c 1)J(b 2 , c 2) and J(a, 1) = ε(a) = J(1, a) (1) for all a, b, c ∈ A. Given a Hopf 2−cocycle J for A, one can construct a new Hopf algebra (A J , m J , 1, ∆, ε, S J) as follows. As a coalgebra, A J = A. The new multiplication is given by m J (a ⊗ b) = J −1 (a 1 , b 1)a 2 b 2 J(a 3 , b 3) (2) for all a, b ∈ A. The new antipode is given by S J (a) = J −1 (a 1 , S(a 2))S(a 3)J(S(a 4), a 5) (3)