## Ternary “Lie structure”

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One of the motivation of the theory of Lie Algebras is that every associative algebra $A$ is a LA when the bracket is defined by $[a,b]=ab-ba$ : this is skew-symmetric and satisfies the Jacobi identity $[[a,b],c]+[[b,c],a]+[[c,a],b]=0$. Conversely, every abstract LA can be embedded into an associative algebra (its envelopping algebra). And for some good reason, one is really interested in sub-LAs rather than sub-algebras. A similar attitude, with different motivation lead to the notion of Jordan algebras.

If $A$ is an associative algebra, one may consider instead the ternary bracket $$[a,b,c]_3=abc+bca+cab-acb-cba-bac.$$ Does $[.,.,.]_3$ satisfy non-trivial identities, besides skew-symmetry? Is there any theory of abstract objects, vector spaces endowed which a ternary skew-symmetric product satisfying these identities?

More generally, we may consider a $d$-bracket, which bears the name of standard non-commutative polynomial in $d$ non-commuting variables. For $d=2$, it is nothing but the standard bracket. When $d=2p$, the $d$-bracket does satisfy non-trivial identies, for instance $$\sum_{i\in\frak A_7}[[a_{i_1},a_{i_2},a_{i_3},a_{i_4}],a_{i_5},a_{i_6},a_{i_7}]=0,\qquad\forall a_1,\ldots,a_7\in A.$$ I don't know if something non-trivial exists when $d$ is odd.

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Have you checked if these are $L_\infty$ structures? – Jim Conant May 10 2012 at 16:50
(Ginzburg and Berger have studied a family of algebras, called antisymmetrizer algebras. The thirds one is the quotient of the free algebra on three generators $a$, $b$, $c$ modulo what you write as $[abc]_3$; the algebra is encodes then properties of triads of elements which "ternary-Lie-commute". The algebra is $3$-Koszul in the sense of Berger, so quite special: maybe you can extract information from that fact) – Mariano Suárez-Alvarez May 10 2012 at 18:21
If you remove $bca-acb$ (that is, the only two terms with $c$ in the middle) then you get a Lie triple system jstor.org/discover/10.2307/… en.wikipedia.org/wiki/… – Fernando Muro May 10 2012 at 19:40
You might be interested in my answer to a previous MO question concerning n-Lie algebras: mathoverflow.net/questions/49437/… – José Figueroa-O'Farrill May 12 2012 at 9:54

## 4 Answers

"Identities for the ternary commutator" by Bremner classifies all such identities up to degree 7. A recent exposition can also be found in "Ternutator Identities" by C. Devchand, D. Fairlie, J. Nuyts, G. Weingart. Similar identities for n-ary commutators are proven in "Multi-operator brackets acting thrice" by T. Curtright, X. Jin, L. Mezincescu. I don't know if there is an accepted definition for what a Lie n-ary algebra should be for $n\geq 3$ (not to be confused with Lie n-algebras from nlab). For $n=3$ the most standard object one encounters are Lie triple systems.

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Ternutators is a painful neologism... – Mariano Suárez-Alvarez May 10 2012 at 18:17
Thanks! This is definitely the answer I was looking for. I'm impressed by the complexity of such a short paper. Page 618 is impressive! The referee must have had hard time... – Denis Serre May 10 2012 at 20:10

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"Ternutation" = the act of neezing. Compare "Sternutation". – Noam D. Elkies May 11 2012 at 2:06
Which gives éternuer in French. – Chandan Singh Dalawat May 11 2012 at 9:29

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The proper setting is the theory of operads, which allows to deal with any number of generators at a single stroke.

The Lie operad is the sub-operad of the associative operad generated by 12-21.

There exists some n-ary Lie operads, but I do not remember whether they are exactly what you ask.

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One way to define a Lie structure on a vector space $V$, is as a map $\wedge^2V\to V$ such that its natural extension to $d\colon\wedge^k V\to \wedge ^{k-1}V$ satisfies $d^2=0$. This exactly gives the Jacobi identity. Similarly one can define a Lie infinity structure to be any map $d\colon \wedge V\to \wedge V$ with square $0$. In your case $[\cdot,\cdot,\cdot]_3$ gives a map $\wedge^k V\to\wedge^{k-2}V$ and the question becomes whether it squares to $0$, which I haven't had time to work out. But if it does, then you have a L-infinity structure.

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You might want to look at the paper "On Lie k-Algebras" by P. Hanlon by M. Wachs (http://www.sciencedirect.com/science/article/pii/S0001870885710389). They consider algebras satisfying the generalized Jacobi identity you specify. I wanted to leave this as a comment but I don't have enough reputation.

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