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  = 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  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  for multiplier Hopf algebras. The first is that the Drinfel'd double  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  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  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.

Key Words:

• Drinfel'd double
• (Group-cograded) Multiplier Hopf algebra
• Hopf G-(co)algebra
• Long module algebra
• Twisted tensor product algebra
• Twisting

Mathematics Subject Classification:

• Primary 16W30
• Secondary 16W50

ACKNOWLEDGMENTS

The authors would like to thank the referee for his/her helpful comments on this article. The second author was supported by the Research Council of the K. U. Leuven. He is very grateful to Professor J. Quaegebeur for his help and to the research group of the K. U. Leuven for providing a good atmosphere to work. In particular, he would like to thank Professor L. Delvaux for many discussions on this topic. This work was partially supported by the FNS of CHINA (10571026), Jiangsu Province (BK2005207).

Notes

Communicated by M. Cohen.