Central invariants and enveloping algebras of braided Hom-Lie algebras

Abstract

Let (H,?) be a monoidal Hom-Hopf algebra and HH HYD the Hom-Yetter-Drinfeld category over (H,?). Then in this paper, we first introduce the definition of braided Hom-Lie algebras and show that each monoidal Hom-algebra in HH HYD gives rise to a braided Hom-Lie algebra. Second, we prove that if (A,?) is a sum of two H-commutative monoidal Hom-subalgebras, then the commutator Hom-ideal [A,A] of A is nilpotent. Also, we study the central invariant of braided Hom-Lie algebras as a generalization of generalized Lie algebras. Finally, we obtain a construction of the enveloping algebras of braided Hom-Lie algebras and show that the enveloping algebras are H-cocommutative Hom-Hopf algebras.