On integrals and cointegrals for quasi-Hopf algebras (original) (raw)

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

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.

Co-Frobenius Hopf Algebras: Integrals, Doi-Koppinen Modules and Injective Objects* 1

Journal of Algebra, 1999

On cotriangular Hopf algebras

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)

COMBINATORIAL HOPF ALGEBRAS IN QUANTUM FIELD THEORY I

Reviews in Mathematical Physics, 2005

This manuscript stands at the interface between combinatorial Hopf algebra theory and renormalization theory. Its plan is as follows: Section 1 is the introduction, and contains as well an elementary invitation to the subject. The rest of part I, comprising Sections 2-6, is devoted to the basics of Hopf algebra theory and examples, in ascending level of complexity. Part II turns around the all-important Faà di Bruno Hopf algebra. Section 7 contains a first, direct approach to it. Section 8 gives applications of the Faà di Bruno algebra to quantum field theory and Lagrange reversion. Section 9 rederives the related Connes-Moscovici algebras. In Part III we turn to the Connes-Kreimer Hopf algebras of Feynman graphs and, more generally, to incidence bialgebras. In Section 10 we describe the first. Then in Section 11 we give a simple derivation of (the properly combinatorial part of) Zimmermann's cancellation-free method, in its original diagrammatic form. In Section 12 general incidence algebras are introduced, and the Faà di Bruno bialgebras are described as incidence bialgebras. In Section 13, deeper lore on Rota's incidence algebras allows us to reinterpret Connes-Kreimer algebras in terms of distributive lattices. Next, the general algebraic-combinatorial proof of the cancellation-free formula for antipodes is ascertained; this is the heart of the paper. The structure results for commutative Hopf algebras are found in Sections 14 and 15. An outlook section very briefly reviews the coalgebraic aspects of quantization and the Rota-Baxter map in renormalization.

Radford's formula for co-Frobenius Hopf algebras

Journal of Algebra, 2007

This note extends Radford's formula for the fourth power of the antipode of a finite dimensional Hopf algebra to co-Frobenius Hopf algebras and studies equivalent conditions to a Hopf algebra being involutory for finite dimensional and co-Frobenius Hopf algebras.

A class of semisimple Hopf algebras acting on quantum polynomial algebras

We construct a class of non-commutative, non-cocommutative, semisimple Hopf algebras of dimension 2n 2 and present conditions to define an inner faithful action of these Hopf algebras on quantum polynomial algebras, providing, in this way, more examples of semisimple Hopf actions which do not factor through group actions. Also, under certain condition, we classify the inner faithful Hopf actions of the Kac-Paljutkin Hopf algebra of dimension 8, H8, on the quantum plane.