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arXiv · 1902.06252

Central invariants and enveloping algebras of braided Hom-Lie algebras

Shengxiang Wang·Xiaohui Zhang·Shuangjian Guo
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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.

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Shengxiang Wang, Xiaohui Zhang, Shuangjian Guo. 2019-02-17. Central invariants and enveloping algebras of braided Hom-Lie algebras. https://arxiv.org/abs/1902.06252

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