(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 (H) over tilde ((HYD)-Y-H(Z)), where m, n is an element of 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 (H) over tilde ((HYD)-Y-H(Z))), and then prove that (A, alpha) can give rise to an (m, n)-Hom-Lie algebra with suitable Lie bracket when the braiding T in (H) over tilde ((HYD)-Y-H(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.