Fundamental Theorems of Weak Doi–Hopf Modules and Semisimple Weak Smash Product Hopf Algebras

Abstract

This paper, mainly gives a Fundamental Theorem of weak Doi–Hopf modules, which is not only generalizes the Fundamental Theorem of weak Hopf modules but also generalizes the Fundamental Theorem of relative Hopf modules. Moreover, it gives a sufficient and necessary condition for weak smash product algebras to be weak bialgebras, and a sufficient condition for weak smash product algebras to be semisimple weak Hopf algebras.

Key Words:

• Weak Hopf algebra
• Weak Doi–Hopf module
• Fundamental Theorem
• Weak smash product algebra

Mathematics Subject Classification:

• 16W30

Acknowledgments

The first author, Zhang Liangyun takes this opportunity to express his thanks to the referee for his or her helpful comments, and thanks to Professor Ding Nan-ging for his help. At the same time, he takes this opportunity to express his thanks to Professor Nicolás Andruskiewitsch for inviting him to visit Ciudad University in Argentina. This work is supported by NSF of China and by the excellent doctorate fund of Nanjing Agricultural University.

Notes

^#Communicated by M. Cohen.