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:


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

  • 4
    $\begingroup$ Do you have any examples you're interested in? $\endgroup$ Feb 21, 2019 at 9:42
  • 3
    $\begingroup$ 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

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.