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

TL;DR

This paper extends the double extension method to classify commutative n-ary superalgebras with skew-symmetric invariant forms using derived bracket formalism, unifying various algebra decomposition techniques.

Contribution

It introduces a unified approach to decompose commutative n-ary superalgebras with invariant forms via derived brackets, generalizing existing methods.

Findings

Any such superalgebra can be constructed through orthogonal sums and generalized double extensions.

The approach unifies decomposition techniques across different algebraic structures.

Provides a systematic inductive construction method for these superalgebras.

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.