Double extension for commutative $n$-ary superalgebras with a skew-symmetric invariant form

Elizaveta Vishnyakova

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arXiv · 1611.09412

Double extension for commutative $n$-ary superalgebras with a skew-symmetric invariant form

Elizaveta Vishnyakova

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Abstract

The method of double extension, introduced by A.~Medina and Ph.~Revoy, is a procedure which decomposes a Lie algebra with an invariant symmetric form into elementary pieces. Such decompositions were developed for other algebras, for instance for Lie superalgebras and associative algebras, Filippov $n$-algebras and Jordan algebras. The aim of this note is to find a unified approach to such decompositions using the derived bracket formalism. More precisely, we show that any commutative $n$-ary superalgebra with a skew-symmetric invariant form can be obtained inductively by taking orthogonal sums and generalized double extensions.

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Elizaveta Vishnyakova. 2016-11-28. Double extension for commutative $n$-ary superalgebras with a skew-symmetric invariant form. https://arxiv.org/abs/1611.09412

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