ON THE GENERALIZED H-LIE STRUCTURE OF ASSOCIATIVE ALGEBRAS IN YETTER-DRINFELD CATEGORIES

ABSTRACT

We study the structure of the generalized H -Lie algebras (i.e., the Lie algebras in the Yetter-Drinfeld category ) for any Hopf algebras and the H -Lie structure of an algebra A in . Let H be arbitrary Hopf algebra. Firstly, We show that if A is a sum of two H -commutative subalgebras, then the H -commutator ideal of A is nilpotent, generalizing the results from [1] for a cotriangular Hopf algebra to the case of any Hopf algebra. Secondly, We investigate the H -Lie ideal structure of A by showing that if A is H -simple, then any non-commutative H -Lie ideal I of A must contain , giving a positive answer to the question given in [[1], p. 42]. Finally, a partial analog of [7] is shown in a more general Hopf algebra setting.

ACKNOWLEDGMENT

The author is grateful to Prof. Xu yong-hua for fruitful discussions. In special, he would like to thank the referee for many helpful comments and corrections. This research is Supported by a grant of NSF of China and also a grant awarded to him by NSF of Henan Provice, China.