Semidirect products of weak multiplier Hopf algebras: Smash products and smash coproducts

ABSTRACT

In this paper, we will develop the smash product of weak multiplier Hopf algebras unifying the cases of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras. We will show that the smash product R#A has a regular weak multiplier Hopf algebra structure if R and A are regular weak multiplier Hopf algebras. We shall investigate integrals on R#A. We also consider the result in the ∗-situation and new examples. Dually, we consider the smash coproduct of weak multiplier Hopf algebras under an appropriate form and integrals on the smash coproduct and we obtain results in the ∗-situation.

KEYWORDS:

• Actions
• smash products
• weak multiplier Hopf algebra

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16T05
• 16S40

Acknowledgments

The authors are very thankful to Professor A. Van Daele for his valuable comments on this paper. In particular, the authors are very grateful to the anonymous referee for his/her thorough review of this work and his/her comments and suggestions which help to improve the first version of this paper.

Additional information

Funding

The work is supported by the NNSFS of China (No. 11371088, No. 11571173, No. 11601231) and the NSF of Jiangsu Province (No. BK20171348).