Enveloping Actions for Partial Hopf Actions

Abstract

Motivated by partial group actions on unital algebras, in this article we extend many results obtained by Exel and Dokuchaev to the context of partial actions of Hopf algebras, according to Caenepeel and Jansen. First, we generalize the theorem about the existence of an enveloping action, also known as the globalization theorem. Second, we construct a Morita context between the partial smash product and the smash product related to the enveloping action. Third, we dualize the globalization theorem to partial coactions and finally, we define partial representations of Hopf algebras and show some results relating partial actions and partial representations.

Key Words:

• Partial coactions
• Partial group actions
• Partial Hopf actions
• Partial representations
• Smash products

2000 Mathematics Subject Classification:

• Primary 16W30
• Secondary 16S40, 16S35, 58E40

ACKNOWLEDGMENTS

The authors would like to thank to Edson R. Álvares and Eduardo O. C. Hoefel for fruitful discussions. The first author (M.M.S.A.) would like to thank Virgínia S. Rodrigues for her fundamental role in establishing the UFPR-UFSC Hopf Seminars. The second author (E.B.) would like to thank the Math Department of UFPR and its staff for their kind hospitality.

Notes

Communicated by M. Cohen.