Central invariants and enveloping algebras of braided Hom-Lie algebras

Abstract

Let $(H,\alpha)$ be a monoidal Hom-Hopf algebra
and $^{H}_{H}\mathcal{HYD}$ the Hom-Yetter-Drinfeld category over $(H,\alpha)$.
Then in this paper, we first introduce the definition of braided Hom-Lie algebras
and show that each monoidal Hom-algebra in $^{H}_{H}\mathcal{HYD}$ gives rise to a braided Hom-Lie algebra.
Second, we prove that if $(A,\beta)$ 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 algerbas.