I was wondering if some concepts of Lie algebras with two multiplications exist in the litterature. By this I mean an object $(A,+, \times , \star)$ such that $(A,+,\times)$ and $(A,+,\star)$ are Lie algebras, and there exist some equationnal affinities between the two products (probably something which looks like the Jacobi identity).
For example : $x\times(y \star z) =(x \times y) \star z + y \star ( x\ \times z) $.
Does anybody has ever heard or read about something similar ?
Post-Lie algebras and post-Lie algebra structures have two Lie algebra structures on a vector space $V$, say a Lie bracket $[x,y]$ and a Lie bracket $\{x,y\}$, together with a bilinear product $x\cdot y$ such that
\begin{align} x\cdot y -y\cdot x & = [x,y]-\{x,y\} \label{post5}\\ [x,y]\cdot z & = x\cdot (y\cdot z) -y\cdot (x\cdot z) \label{post6}\\ x\cdot \{y,z\} & = \{x\cdot y,z\}+\{y,x\cdot z\} \label{post7} \end{align} for all $x,y,z \in V$. This is related to geometric structures on Lie groups, Rota-Bater operators, homology of partition posets, Koszul operads, isospectral flows, Lie-Butcher series and many other topics. For a reference see for example the articles Post-Lie algebra structures on pairs of Lie algebras and Rota-Baxter operators and post-Lie algebra structures on semisimple Lie algebras.