On the Twisting and Drinfel'd Double for Multiplier Hopf Algebras

Abstract

Let ⟨ A, B⟩ be a pairing of two regular multiplier Hopf algebras A and B. One method of constructing the Drinfel'd double 𝒟 = A⋈ B cop is by the use of an invertible twist map R: B⊗ A→ A⊗ B defining an associative product on A⊗ B. In Delvaux (2003) and Drabant and Van Daele (2001), the authors construct R by , where R 2 and is only related to the module actions between A and B. Another way is given in Delvaux and Van Daele (2004a) in which the authors also just consider the module actions and then construct the Drinfel'd double 𝒟 as an algebra of operators on the vector space B⊗ A. In this article we will give two different points of view of constructing the Drinfel'd double 𝒟 for multiplier Hopf algebras. The first is that the Drinfel'd double 𝒟 associated to the pairing ⟨ A, B⟩ is constructed by using not only the module actions but also the comodule coactions, i.e., the Drinfel'd double 𝒟 is given in the framework of a special twisted tensor product algebra structure A ⊡ A op ⊗ A B. The second is as follows: If P is a multiplier Hopf algebra and a reduced (A, B)-bicomodule algebra (an A-Long module algebra), then we present a twisting construction of the product of P via the coactions of A and B on P, and we show that the Drinfel'd double 𝒟 is isomorphic as a multiplier Hopf algebra to the opposite twisting of (A op, cop ⊗ B op, cop ). As an application of our theory, we consider the case of group-cograded multiplier Hopf algebras and the case of Hopf group-coalgebras.