Elizaveta Vishnyakova

On n-ary analogue of Lie (super)algebras

A different reading of the standard Jacobi identity leads to various generalizations of the notion of Lie superalgebra for $n$-ary case. The most popular n-ary analogues were suggested by V.T.Filippov, P.Michor, A.Vinogradov, M.Vinogradov and other. For instance, A.Vinogradov and M.Vinogradov introduced a two parameter series of n-ary Lie superalgebras. The interesting fact here is that this series contains also commutative associative algebras.

We will discuss the following: this theory in the context of quadratic n-ary Lie superalgebras using a "derived bracket" approach from Poisson Geometry; classification of simple n-ary Lie algebras, a decomposition of such algebras into elementary pieces.