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The Lie operad $\text{Lie}$ is generated by a binary operator $[\ ,\ ]$, modulo a degree two relation (skew commutativity $[x,y]=-[y,x]$) and a degree three relation (Jacobi $[x,[y,z]]+[y,[z,x]]+[z,[x,y]]$). See e.g. the paper "The operad Lie is free".

Question: do "non(skew)commutative Lie algebras" ever show up in nature? i.e. is there an operad $\text{Lie}^{\mathrm{nc}}$ generated by a binary operator $[\ ,\ ]$ subject to Jacobi and some subset of the relations generated by Jacobi and skew-commutativity (so there is a map $\text{Lie}^{\mathrm{nc}}\to \text{Lie}$) but not including skew-commutativity, and so that its algebras $\text{Alg}_{\text{Lie}^{\mathrm{nc}}}$ show up in e.g. algebra or geometry?

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    $\begingroup$ This is not what you are asking for, but an interesting and sort of related question goes the other way, natural maps $\mathrm{Lie}\to \mathrm{Lie}^{\mathrm{nc-ext}}$ where the target is generated by a single not-necessarily commutative binary operation and the inclusion of generators is as the commutator. The most famous such extension is the associative operad, but you can also look up PreLie and Lie-admissible. $\endgroup$ Commented Mar 25, 2023 at 19:29

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Such objects are known as Leibniz algebras. A Leibniz algebra is a module $M$ together with a bilinear pairing $$[-,-]\colon M⊗M→M$$ that satisfies the Leibniz identity: $$[a,[b,c]]=[[a,b],c]+[b,[a,c]].$$

Leibniz algebras for which $[a,b]=-[b,a]$ are precisely Lie algebras.

The concept was introduced and studied by A. Blokh: A generalization of the concept of a Lie algebra.

Later, it was studied by Loday, who also coined the name “Lebniz algebra”: Une version non commutative des algèbres de Lie: les algèbres de Leibniz.

The article Leibniz algebra contains many more additional references.

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  • $\begingroup$ Thanks. Are you also making the claim that there are (probably) no natural intermediate notions between Lie and Leibniz algebras (i.e. operads with more relations than Leibniz but fewer than Lie)? $\endgroup$
    – Pulcinella
    Commented Mar 25, 2023 at 17:22
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    $\begingroup$ It is often (but by no means always) the case that the relations on operads we think of as "natural" are either (1) symmetry relations on the generators, (2) quadratic in the generators, or (3) unitality relations. There are no intermediate notions satisfying these three conditions. $\endgroup$ Commented Mar 25, 2023 at 18:13
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    $\begingroup$ I guess the answer is "read the references", but, of the many ways of re-writing the Jacobi identity that are equivalent in the presence of skew-commutativity, why is the Leibniz identity the "best" one? $\endgroup$
    – LSpice
    Commented Mar 25, 2023 at 18:14
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    $\begingroup$ My comment is wrong, one intermediate notion satisfying my criteria is "Leibniz algebras having $[a,b]$ skew-central for all $a,b$." $\endgroup$ Commented Mar 25, 2023 at 18:31
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    $\begingroup$ @Pulcinella: Gabe just gave one intermediate example in his comment. The point is that such examples will be typically considered in the same literature as Leibniz algebras. $\endgroup$ Commented Mar 25, 2023 at 18:59

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