On integrals and cointegrals for quasi-Hopf algebras (original) (raw)
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Integrals for (dual) quasi-Hopf algebras. Applications
Journal of Algebra, 2003
A classical result in the theory of Hopf algebras concerns the uniqueness and existence of integrals: for an arbitrary Hopf algebra, the integral space has dimension ≤ 1, and for a finite dimensional Hopf algebra, this dimension is exaclty one. We generalize these results to quasi-Hopf algebras and dual quasi-Hopf algebras. In particular, it will follow that the bijectivity of the antipode follows from the other axioms of a finite dimensional quasi-Hopf algebra. We give a new version of the Fundamental Theorem for quasi-Hopf algebras. We show that a dual quasi-Hopf algebra is co-Frobenius if and only if it has a non-zero integral. In this case, the space of left or right integrals has dimension one.
An Approach to Hopf Algebras via Frobenius Coordinates I
Beitrage zur Algebra und Geometrie
In Section 1 we introduce Frobenius coordinates in the general setting that includes Hopf subalgebras. In Sections 2 and 3 we review briefly the theories of Frobenius algebras and augmented Frobenius algebras with some new material in Section 3. In Section 4 we study the Frobenius structure of an FH-algebra H [25] and extend two recent theorems in . We obtain two Radford formulas for the antipode in H and generalize in Section 7 the results on its order in . We study the Frobenius structure on an FH-subalgebra pair in Sections 5 and 6. In Section 8 we show that the quantum double of H is symmetric and unimodular. MSC 2000: 16W30 (primary); 16L60 (secondary)
An approach to Hopf algebras via Frobenius coordinates II
Journal of Pure and Applied Algebra, 2002
In Section 1 we introduce Frobenius coordinates in the general setting that includes Hopf subalgebras. In Sections 2 and 3 we review briefly the theories of Frobenius algebras and augmented Frobenius algebras with some new material in Section 3. In Section 4 we study the Frobenius structure of an FH-algebra H [25] and extend two recent theorems in . We obtain two Radford formulas for the antipode in H and generalize in Section 7 the results on its order in . We study the Frobenius structure on an FH-subalgebra pair in Sections 5 and 6. In Section 8 we show that the quantum double of H is symmetric and unimodular. MSC 2000: 16W30 (primary); 16L60 (secondary)
The Quantum Double for Quasitriangular Quasi-Hopf Algebras
Communications in Algebra, 2003
Let D(H) be the quantum double associated to a finite dimensional quasi-Hopf algebra H, as in [9] and [10]. In this note, we first generalize a result of Majid [15] for Hopf algebras, and then prove that the quantum double of a finite dimensional quasitriangular quasi-Hopf algebra is a biproduct in the sense of [4]. * Research supported by the bilateral project "Hopf Algebras in Algebra, Topology, Geometry and Physics" of the Flemish and Romanian governments. † This paper was written while the first author was visiting the Free University of Brussels, VUB (Belgium); he would like to thank VUB for its warm hospitality.
Integral Theory for Quasi-Hopf Algebras
1999
We generalize the fundamental structure Theorem on Hopf (bi)-modules by Larson and Sweedler to quasi-Hopf algebras H. If H is finite dimensional this proves the existence and uniqueness (up to scalar multiples) of integrals in H. Among other applications we prove a Maschke type Theorem for diagonal crossed products as constructed by the authors.
Quantum double for quasi-Hopf algebras
1998
We introduce a quantum double quasitriangular quasi-Hopf algebra D(H) associated to any quasi-Hopf algebra H. The algebra structure is a cocycle double cross product. We use categorical reconstruction methods. As an example, we recover the quasi-Hopf algebra of Dijkgraaf, Pasquier and Roche as the quantum double D φ (G) associated to a finite group G and group 3-cocycle φ. Keywords: quantum double-quasi-Hopf algebra-finite group-cocycle-category-reconstruction.
Co-Frobenius Hopf Algebras: Integrals, Doi-Koppinen Modules and Injective Objects* 1
Journal of Algebra, 1999
We investigate Hopf algebras with non-zero integral from a coalgebraic point of view. Categories of Doi-Koppinen modules are studied in the special case where the defining coalgebra is left and right semiperfect, and several pairs of adjoint functors are constructed. As applications we give a very short proof for the uniqueness of the integrals and provide information about injective objects in the category of Doi-Koppinen modules.
American Journal of Mathematics, 2001
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)