# Weak associativity

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?

• Do you have any examples you're interested in? Feb 21, 2019 at 9:42
• 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". Feb 21, 2019 at 13:52

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.