A Lot of Quasitriangular Group-cograded Multiplier Hopf Algebras

Abstract

A new method of constructing quasitriangular group-cograded multiplier Hopf algebras is provided. For a multiplier Hopf dual pairing σ between regular multiplier Hopf algebras A and B, we introduce the concept of a σ -compatible pairing ( , , σ ), where and are actions of the twisted semi-direct group S (G) of a group G on A and B, respectively. We construct a twisted double group-cograded multiplier Hopf algbera D (A, B; σ, , ). Furthermore, if there is a canonical multiplier in M(B ⊗ A) we show existence of quasitriangular structure on D (A, B; σ, , ). As an application, special cases and examples are given.

A Class of Quasitriangular Group-Cograded Multiplier Hopf Algebras

“…Then we have the main results of this section: there exists a multiplier Hopf algebra structure on D(A, B), which generalizes the classical construction of finite dimensional Hopf algebras by Panaite and Staic Mihai in [7]. This construction is different from what introduced in [12]. • For any (α, β) ∈ G, the multiplication of A ⊲⊳ B (α,β) is given by Definition 3.1.…”

Heisenberg double as braided commutative Yetter–Drinfel’d module algebra over Drinfel’d double in multiplier Hopf algebra case

“…Let i ∈ N, the set of natural integers and λ ∈ C such that λ i is a primitive mth root of 1. Then we recall from [14] that the Hopf algebra B is the algebra with generators c and X satisfying relations: cX = λXc and X m = 0. The Hopf algebra structure on B is given by [14](only need to let α = β = γ = δ = ι), then we can get H is a D-Yetter-Drinfel'd module algebra, and it is braided commutative.…”

“…denote S A (a) just as S(a). Drinfel'd double D can be considered as a special case of twisted double defined in [14].…”

“…Let B be a (infinite dimensional) co-Frobenius Hopf algebra with a left integral ϕ, and A be the dual multiplier Hopf algebra shown in [17]. Then by [6] or Corollary 3.6 in [14], let Φ α = ι = Φ β , we can get the Drinfel'd double D = A ⊲⊳ B with structures given by the following formulas: Example 4.2 Let C be an infinite cyclic group with generator c and let m be a positive integer. Let i ∈ N, the set of natural integers and λ ∈ C such that λ i is a primitive mth root of 1.…”

“…Multiplier Hopf algebras, considered as a generalization of Hopf algebras, play a very important role in the duality of a class of infinite dimensional Hopf algebras (see [11,12]). Many constructions, such as Drinfel'd double (see [6,14]), Yetter-Drinfel'd module (see [5,15]), have naturally generalized. Multiplier Hopf algebra becomes a useful tool to deal with some infinite dimensional Hopf algebra questions.…”