Group-cograded Multiplier Hopf ${\left( { * {\text{ - }}} \right)}$ algebras

Abstract

Let G be a group and assume that ( A p ) p∈G is a family of algebras with identity. We have a Hopf G-coalgebra (in the sense of Turaev) if, for each pair p, q ∈ G, there is given a unital homomorphism p,q : A pq → A p ⊗ A q satisfying certain properties. Consider now the direct sum A of these algebras. It is an algebra, without identity, except when G is a finite group, but the product is non-degenerate. The maps p,q can be used to define a coproduct on A and the conditions imposed on these maps give that ( A, ) is a multiplier Hopf algebra. It is G-cograded as explained in this paper. We study these so-called group-cograded multiplier Hopf algebras. They are, as explained above, more general than the Hopf group-coalgebras as introduced by Turaev. Moreover, our point of view makes it possible to use results and techniques from the theory of multiplier Hopf algebras in the study of Hopf group-coalgebras (and generalizations). In a separate paper, we treat the quantum double in this context and we recover, in a simple and natural way (and generalize) results obtained by Zunino. In this paper, we study integrals, in general and in the case where the components are finite-dimensional. Using these ideas, we obtain most of the results of Virelizier on this subject and consider them in the framework of multiplier Hopf algebras.

A Lot of Quasitriangular Group-cograded Multiplier Hopf Algebras

“…We first recall from [5] that a multiplier Hopf dual pairing between A and B is a linear map σ ∈ Hom(A ⊗ B, K) such that, σ (a, bb ) = σ (a (1) , b )σ (a (2) , b ),…”

“…For a ∈ A, b ∈ B, we can define a b , b a, a b and b a in the following way. For a ∈ A and b ∈ B, we have: (b a)a = σ (a (2) (1) , b )a (2) a and (b a)b = σ (a, b (1) )b (2) b . The regularity conditions on the dual paring σ say that the multipliers b a and a b in M(A) (resp.…”

“…for all a, a ∈ A, b , b ∈ B and α, β ∈ G. Proof We define linear maps T 1 and T 2 by the formulas: (1) . Then T 1 and T 2 are bijective on D(A, B; σ, φ, , (α, β)) with the inverses given by…”

“…By Theorem 2.6, we have D(A, B; σ, , ) is a Hopf S (G)-coalgebra with a product, a comultiplication , a counit ε, and an antipode S given by the following formulas: (1) ), S −1 B (b (3) ))(aa (2) ⊗ b (2) b )σ ( β (a (3) ), b (1) ), (α,β),(λ,γ ) (a ⊗ b ) = (a (2) ⊗ γ −1 (b (1) )) ⊗ (a (1) …”

“…Recently, the concept of a group-cograded multiplier Hopf algebra was introduced by Abd El-hafez, Delvaux and Van Daele in [1] as a generalization of Hopf group-coalgebras introduced in [10]. Let (A, ) be a multiplier Hopf algebra and G a group.…”