Smash Products of H-Simple Module Algebras

Abstract

Let H be a finite-dimensional Hopf algebra and A a finite-dimensional H-simple left H-module algebra. We show that the smash product A#H is isomorphic to End _A′(V ⊗ H*), where V ≠ 0 is a finite-dimensional left A-module and (A′, V′) the stabilizer of (A, V). As an application it is proved that A#H is isomorphic to a full matrix algebra over A′ when H is semisimple and dim V|dim A.

Key Words:

• Full matrix algebra
• Hopf algebra
• H-simple module algebra
• Smash product

2010 Mathematic Subject Classification:

• 16W30

ACKNOWLEDGMENT

The authors would like to thank the referee for his/her careful reading of the paper and helpful comments.

This work is supported by National Natural Science Foundation of China (10731070 and 11126121) and Project of Science and Technology Department of Henan Province of China (102300410233).

Notes

Communicated by M. Cohen.