# Central invariants and enveloping algebras of braided Hom-Lie algebras

## TL;DR

This paper introduces braided Hom-Lie algebras within a monoidal Hom-Hopf algebra framework, explores their central invariants, and constructs their enveloping algebras, extending classical Lie theory to a braided Hom setting.

## Contribution

It defines braided Hom-Lie algebras, links monoidal Hom-algebras to these structures, and constructs their enveloping algebras as $H$-cocommutative Hom-Hopf algebras.

## Key findings

- Braided Hom-Lie algebras are derived from monoidal Hom-algebras in the Hom-Yetter-Drinfeld category.
- The commutator of a sum of two $H$-commutative subalgebras is nilpotent.
- Enveloping algebras of braided Hom-Lie algebras are $H$-cocommutative Hom-Hopf 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.