Nonlinear realisations of Lie superalgebras

Abstract

For any decomposition of a Lie superalgebra $\mathcal{G}$ into a direct sum $\mathcal{G}=\mathcal{H}\oplus \mathcal{E}$ of a subalgebra $\mathcal{H}$ and a subspace $\mathcal{E}$, without any further resctrictions on $\mathcal{H}$ and $\mathcal{E}$, we construct a nonlinear realisation of $\mathcal{G}$ on $\mathcal{E}$. The result generalises a theorem by Kantor from Lie algebras to Lie superalgebras. When $\mathcal{G}$ is a differential graded Lie algebra, we show that it gives a construction of an associated $L_{\infty }$-algebra.

arXiv commentv2: 26 pages, several minor changes. v3: Version accepted for publication in Communications in Algebra

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