On the generalized Lie structure of associative algebras
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We study the structure of Lie algebras in the categoryHMA ofH-comodules for a cotriangular bialgebra (H, 〈|〉) and in particular theH-Lie structure of an algebraA inHMA. We show that ifA is a sum of twoH-commutative subrings, then theH-commutator ideal ofA is nilpotent; thus ifA is also semiprime,A isH-commutative. We show an analogous result for arbitraryH-Lie algebras whenH is cocommutative. We next discuss theH-Lie ideal structure ofA. We show that ifA isH-simple andH is cocommutative, then any non-commutativeH-Lie idealU ofA must contain [A, A]. IfU is commutative andH is a group algebra, we show thatU is in the graded center ifA is a graded domain.
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