Group Twisted Smash Products and Doi–Hopf Modules for T-Coalgebras

Abstract

The notions of Doi–Hopf group module and group twisted smash product are defined as a respective generalization of an ordinary Doi–Hopf module and a usual twisted smash product. It is shown that the Zunino's Yetter-Drinfel'd modules are special cases as these new Doi–Hopf group modules and that the Zunino's Drinfel'd double appears as a type of such a group twisted smash product, respectively. Furthermore, the concepts of group skew pair and generalized group smash product are introduced. As for a group skew pair σ for a T- coalgebra H and a T-algebra B, it is proved that a group twisted smash product is a Hopf group coalgebra if H has a bijective antipode, and that is a special case of such a generalized group smash product.

Key Words:

• T-Coalgebra
• Doi–Hopf group module
• Group twisted smash product
• Zunino's Drinfel'd double

Mathematical Subject Classification:

• 16W30
• 16W50

Acknowledgments

The author would like to thank the referee for his/her helpful comments on this paper. This work was supported in part by the Science Foundation of Henan Province for Distinguished Young Scholars and the NSF of China.

Notes

^#Communicated by M. Cohen.