Classification of simple linearly compact n-Lie superalgebras

TL;DR

This paper classifies simple linearly compact n-Lie superalgebras for n>2 over a field of characteristic zero, linking them to specific Z-graded Lie superalgebras and providing a list of four examples.

Contribution

It introduces a classification method for simple linearly compact n-Lie superalgebras based on their correspondence with certain Z-graded Lie superalgebras.

Findings

Four classified examples of simple linearly compact n-Lie superalgebras.

Identification of the n+1-dimensional vector product n-Lie algebra as one example.

Description of three infinite-dimensional n-Lie algebras.

Abstract

We classify simple linearly compact n-Lie superalgebras with n>2 over a field F of characteristic 0. The classification is based on a bijective correspondence between non-abelian n-Lie superalgebras and transitive Z-graded Lie superalgebras of the form L=\oplus_{j=-1}^{n-1} L_j, such that L_{-1}=g, where dim L_{n-1}=1, L_{-1} and L_{n-1} generate L, and [L_j, L_{n-j-1}] =0 for all j, thereby reducing it to the known classification of simple linearly compact Lie superalgebras and their Z-gradings. The list consists of four examples, one of them being the n+1-dimensional vector product n-Lie algebra, and the remaining three infinite-dimensional n-Lie algebras.