# Lie algebras with two multiplications

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 : $$x\times(y \star z) =(x \times y) \star z + y \star ( x\ \times z)$$.

• denoting the null lie bracket $(x\times y) = 0$ and denoting $x\star y$ any other lie bracket that you can define in this same lie algebra, they satisfy this relation. But well, a little bit uninteresting – math.h Jun 12 at 16:59
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.
• We also obtain conditions without the third one $x\cdot y$, which you could consider then, e.g., $\{[x,y],z\}+\{[y,z],x\}+\{[z,x],y\}=0$ for all $x,y,z$. There are several ways to consider triples $(V,[,],\{,\})$. – Dietrich Burde Jun 13 at 12:03