Can Hom-Lie algebras be seen as an $\Omega$-algebras?

Question

An $\Omega$-algebra over a field $K$ is a $K$-algebra $A$ with a set of multilinear operators $\Omega$, where $\Omega=\bigcup_{m=1}^{\infty} \Omega_{m}$ and each $\Omega_{m}$ is a set of $m$-array multilinear operators on $A$. On the other side, let consider the definition of Hom-Lie algebras as follows:

A Hom-algebra $L$ is called a Hom-Lie algebra, if $L$ is anticommutative as an algebra. i.e., $[x,y]=-[y,x]$ and the following identityholds $$ [[x,y],\alpha(z)]+[[z,x],\alpha(y)]+[[y,z],\alpha(x)]=0, $$ for any $x,y,z \in L$ and $\alpha: L \to L$ is a linear map.

Can Hom-Lie algebras be seen as a $\Omega$-algebra?

Answer 1

Yes, Hom-Lie algebras can be considered in the framework of multiple operated algebras. We have finished this paper and we will pose it on Arxiv in the next time. Please see our recent paper Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras

for the relative topies. What's more, the Grobner-Shirshov basis of these algebraic structures have been done by our another paper.