On the generalized Lie structure of associative algebras
- Cite this article as:
- Bahturin, Y., Fischman, D. & Montgomery, S. Israel J. Math. (1996) 96: 27. doi:10.1007/BF02785532
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.