n-aray algebras

A perprint by de Azcarraga and Izquierdo has appeared on the arXiv entitled "n-aray algebras:a review with applications" (arXiv:1005.1028)

In it they review two generalisations of Lie algebra, the Filippov algebras (n-Lie) and the generalised Poisson structures.

The Fulippov algebras, specifically n=3 has found an interesting application in M-theory viz the Bagger-Lambert-Gustavsson model. This has revitalised the interest in these kinds of algebras.

de Azcarraga 's generalised Poisson structures are n-aray generalisations of the classical Poisson structures. Such structures are in fact related to the [math]L_{\infty}[/math]-algebras of Stasheff. This is briefly mentioned in the text.

[math]L_{\infty}[/math]-algebras (with various gradings and symmetry) can be found lying behind the classical BV-antifield formalism, deformation theory and string field theory, for example.

These higher Lie algebras are all things I am very interested in, both from a mathematical and physical view point. Unfortunately, there seems no physical application of the generalised Poisson structures (nor the higher Poisson structures that Th. Voronov initially studied and that I worked on [generalising the Koszul--Schouten bracket and reformulating all this on Lie algebroids]). That said, as mathematics often is just waiting for an application it is very conceivable that the role of Lie theory in physics gets replaced with "higher Lie structures".

Anyway, anyone who is interested in what is "hot" in mathematical physics is advised to take a look. (I can also provide other references if anyone is interested!)

Edited May 12, 201015 yr by Mr Skeptic
added "a" to "algebrs" in title