Classification of Simple Linearly Compact n-Lie Superalgebras

Abstract

We classify simple linearly compact n-Lie superalgebras with n > 2 over a field $${\mathbb{F}}$$ of characteristic 0. The classification is based on a bijective correspondence between non-abelian n-Lie superalgebras and transitive $${\mathbb{Z}}$$ -graded Lie superalgebras of the form $${L=\oplus_{j=-1}^{n-1} L_j}$$ , 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 $${\mathbb{Z}}$$ -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.