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Let $(V,*)$ be an algebra and denote $A_*\in \text{Hom}(V^{\otimes 3},V)$ the associator of the binary product $*\in \text{Hom}(V^{\otimes 2},V)$ defined as $A_*(a,b,c):=(a*b)*c-a*(b*c)$.

The associator $A_*$ is assumed to enjoy the following property:

$A_*(a,b,c)+A_*(b,c,a)-A_*(b,a,c)=0$.

Question: Does this "weak associativity" condition have a name and are there some references discussing it?

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    $\begingroup$ Do you have any examples you're interested in? $\endgroup$ Feb 21, 2019 at 9:42
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    $\begingroup$ Dear Qiaochu, my motivating example is the following: Let me denote $f\cdot g:=f*g+g*f$ the symmetric part of $*$ and $\{f,g\}:=f*g-g*f$ the skewsymmetric part. Then, assuming that the symmetric product $\cdot$ is associative, then $(V,\cdot,\{\cdot,\cdot\})$ is a Poisson algebra if and only if $*$ is "weakly associative". $\endgroup$ Feb 21, 2019 at 13:52

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Let me assume that the characteristic of the ground field is different from two. Let me start by replacing your identity by something where the existing symmetries are a bit more apparent. I claim that your identity for the associator (which I will denote simply $(a,b,c)$, dropping $A_*$) is equivalent to the following two identities: $$ \begin{gather} (a,b,c)+(b,c,a)+(c,a,b)=0,\\ (a,b,c)+(c,b,a)=0. \end{gather} $$ First, they clearly imply your identity: $$0=(a,b,c)+(b,c,a)+(c,a,b)=(a,b,c)+(b,c,a)-(b,a,c).$$ Second, your identity, if we set $a=b=c$, becomes third power associativity $(a,a,a)=0$, and multilinearizing that, we get $$ (a,b,c)+(a,c,b)+(b,c,a)+(b,a,c)+(c,a,b)+(c,b,a)=0. $$ Using your identity $(a,b,c)+(b,c,a)=(b,a,c)$ and the identity $(a,c,b)+(c,b,a)=(c,a,b)$ obtained by acting by the transposition $b\leftrightarrow c$, we obtain the identity $(b,a,c)+(c,a,b)=0$, and we use the same calculation as before: $$0=(a,b,c)+(b,c,a)-(b,a,c)=(a,b,c)+(b,c,a)+(c,a,b).$$ These two identities, in turn, are manifestly equivalent to the system of identities $$ \begin{gather} (a,b,c)+(a,c,b)+(b,c,a)+(b,a,c)+(c,a,b)+(c,b,a)=0,\\ (a,b,c)+(c,b,a)=0. \end{gather} $$ The first of them defines Lie admissible algebras, and the second defines flexible algebras. Search for papers that include "flexible Lie-admissible algebras" in the title brings 15 matches on MathSciNet, including, for instance, Benkart, Georgia M.; Osborn, J. Marshall. Flexible Lie-admissible algebras. J. Algebra 71 (1981), no. 1, 11–31. I understand that this class of algebras was studied because it contains the so called "Okubo algebra" arising in one of the constructions of octonions.

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