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

### Volume 110 / 2016

Banach Center Publications 110 (2016), 287-293 MSC: 17B20, 17B01. DOI: 10.4064/bc110-0-18

#### 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.

#### Authors

• Elizaveta VishnyakovaMax Planck Institute for Mathematics
Bonn
Vivatsgasse, 7
53111 Bonn, Germany
e-mail

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