$(m,n)$-Hom-Lie algebras

Let $(H,\beta)$ be a monoidal Hom-Hopf algebra, and $(A,\alpha)$ an algebra in the $(m,n)$-Hom-Yetter—Drinfeld category $\widetilde{\mathcal{H}}(_H^H{\mathcal{YD(Z)}})$, where $m, n\in\mathcal{Z}$ (the set of integers). In this paper, we introduce the notion of $(m,n)$-Hom-Lie algebra (i.e., Lie algebras in the category $\widetilde{\mathcal{H}}(_H^H{\mathcal{YD(Z)}})$), and then prove that $(A,\alpha)$ can give rise to an $(m,n)$-Hom-Lie algebra with suitable Lie bracket when the braiding $\tau$ in $\widetilde{\mathcal{H}}(_H^H{\mathcal{YD(Z)}})$ is symmetric on $(A,\alpha)$. We also show that if also $(A,\alpha)$ is a sum of two $(H,\beta)$-commutative Hom-subalgebras, then $[A,A][A,A]=0$.