On Cleft Extensions of the Tensor Product of Two Hopf Algebras

Abstract

In this article, we first decompose a cleft extension for X ⊗ Y into two cleft extensions for X and Y respectively, where X and Y are Hopf algebras on a commutative ring R. Conversely, we introduce the concept of consistent cleft Hopf extensions and prove that one can construct a cleft extension for X ⊗ Y by two cleft extensions for X and Y if and only if these two cleft extensions are consistent. An example is also given to show an application of our main results.

Key Words:

• Cleft extensions
• Crossed systems
• Hopf algebras

2010 Mathematics Subject Classification:

• 16T15

ACKNOWLEDGEMENT

The author wish to thank Professor Guoquan Hu for many helpful conversations and professor George Szeto for his valuable advice.

Notes

Communicated by V. A. Artamonov.