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Classification of Simple Linearly Compact n-Lie Superalgebras

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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 nj−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.

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Correspondence to Nicoletta Cantarini.

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Communicated by Y. Kawahigashi

Partially supported by Progetto di ateneo CPDA071244.

Partially supported by an NSF grant.

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Cantarini, N., Kac, V.G. Classification of Simple Linearly Compact n-Lie Superalgebras. Commun. Math. Phys. 298, 833–853 (2010). https://doi.org/10.1007/s00220-010-1049-0

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