Blattner–Cohen–Montgomery's Duality Theorem for (Weak) Group Smash Products

Abstract

We first prove the fundamental theorem for (weak) relative Hopf group-comodules in this article. Secondly, we introduce the notion of a (weak) group smash product and give a sufficient and necessary condition under which (weak) group smash product algebras and the usual tensor product coalgebra become a (weak) semi-Hopf group-coalgebra. Furthermore, we get a sufficient condition for (weak) group smash product algebras to be semisimple. Finally, we prove an analog of the Blattner–Cohen–Montgomery's duality theorem for (weak) group smash products: For any (weak) π-H-module algebra A, there is a canonical isomorphism between the algebras and End(A # H_α)_A.

Key Words:

• Duality theorem
• (Weak) Hopf group-coalgebra
• (Weak) group smash product
• (Weak) relative Hopf group-comodules

2000 Mathematics Subject Classification:

• 16W30

ACKNOWLEDGMENT

The authors would like to thank the referee for his/her valuable comments on this article. This work was partially supported by the Specialized Research Fund for the Doctoral Program of Higher Education (20060286006) and the FNS of CHINA (10571026).

Notes

Communicated by M. Cohen.