We study the structure of the generalized H-Lie algebras (i.e., the Lie algebras in the Yetter-Drinfeld category (HYD)-Y-H) for any Hopf algebras and the H-Lie structure of an algebra A in (HYD)-Y-H. 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 [A, A], 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. [References: 18]
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