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This is a reference request. I came across the class of nonassociative algebras satisfying the following identity: $$ (a,b,c)+\omega(b,c,a)+\omega^2(c,a,b)=0. $$ Here:

  • by an "algebra" I mean a vector space $V$ equipped with a binary operation $V\otimes V\to V$,
  • $(a,b,c)$ denotes the associator $(ab)c-a(bc)$,
  • $\omega$ denotes a primitive third root of unity (which is assumed to exist in the ground field).
I would like to know if precisely this class of algebras was studied in the literature. When I write "precisely", I mean that I do not want classes of algebras where stronger identities hold. (For instance, algebras satisfying this identity and the third power associativity identity $(xx)x=x(xx)$ were studied in a paper of Kleinfeld called "Associator dependent rings".) The reason to insist on absence of extra identities is that I am especially interested in free algebras of this class (or even in the operad describing algebras of this class), although any extra information would help.
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    $\begingroup$ Note that if you call this $\omega$-something, then if $A$ is an $\omega$-something algebra then $A^{\mathrm{op}}$ is $\omega^2$-something. This can justify to keep track of $\omega$ in the terminology (also if Galois conjugation comes into play). $\endgroup$
    – YCor
    Commented Mar 15, 2022 at 22:09
  • $\begingroup$ @YCor first I want to determine if this class deserves to be named ;) $\endgroup$
    – Vladimir Dotsenko
    Commented Mar 17, 2022 at 20:08
  • $\begingroup$ @YCor : this new preprint partially reveals my motivation for the question (though this identity ended up hidden in the parametric family of identities discussed there): arxiv.org/abs/2203.11142 $\endgroup$
    – Vladimir Dotsenko
    Commented Mar 22, 2022 at 5:09

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