Symmetric Pairs in Yetter–Drinfeld Categories over Weak Hopf Algebras

Abstract

In this paper we study the new notion of a symmetric pair of objects in the Yetter–Drinfeld category over a weak Hopf algebra H with a bijective antipode. Then we show that the (co)commutativity and trivial property of H are determined by some symmetric pairs of objects in , generalizing the main results in Cohen and Westreich and Pareigis. Finally, we study the u-condition generalizing the corresponding results appeared in Cohen and Westreich.

Key Words:

• Symmetric pair
• u-condition
• Weak Hopf algebra
• Yetter–Drinfeld category

2010 Mathematics Subject Classification:

• 16T05
• 16T15

ACKNOWLEDGMENTS

The authors would like to thank the referee for her/his comments.

Notes

Communicated by M. Cohen.