Commutative $n$-ary superalgebras with an invariant skew-symmetric form

TL;DR

This paper classifies certain invariant superalgebras, develops Hodge theory for $L_{inite}$-algebras, and explores quasi-Frobenius structures, advancing the understanding of $n$-ary superalgebras with invariant forms.

Contribution

It provides a classification of anti-commutative $m$-dimensional $(m-3)$-ary algebras and real simple Lie $(m-3)$-algebras with positive definite invariant forms, and develops Hodge theory for $L_{inite}$-algebras.

Findings

Classified anti-commutative $(m-3)$-ary algebras with invariant forms.

Classified real simple Lie $(m-3)$-algebras with positive definite invariant forms.

Developed Hodge theory for $L_{inite}$-algebras.

Abstract

We study $n$-ary commutative superalgebras and $L_{\infty}$-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their $n$-ary generalizations, commutative associative and Jordan algebras with an invariant form. We give a classification of anti-commutative $m$-dimensional $(m-3)$-ary algebras with an invariant form, and a classification of real simple $m$-dimensional Lie $(m-3)$-algebras with a positive definite invariant form up to isometry. Furthermore, we develop the Hodge Theory for $L_{\infty}$-algebras with a symmetric invariant form, and we describe quasi-Frobenius structures on skew-symmetric $n$-ary algebras.