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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 ?

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    $\begingroup$ What does motivate this question? $\endgroup$ – Berci Jun 12 at 15:22
  • $\begingroup$ 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 $\endgroup$ – math.h Jun 12 at 16:59
  • $\begingroup$ The motivation is that there exists a categorical description for Lie algebras. So I want to apply the same characterisation to algebras with two multiplications and see what comes out. What I expect is some kind of Lie algebras where the brackets talk to each other. $\endgroup$ – Sov Jun 13 at 12:02
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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.

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  • $\begingroup$ Thank you for the references ! But here it is more likely to have three multiplications, and the third one makes speak the other two together. $\endgroup$ – Sov Jun 13 at 11:59
  • $\begingroup$ 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 $(V,[,],\{,\})$. $\endgroup$ – Dietrich Burde Jun 13 at 12:03

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