Completable nilpotent Lie superalgebras

Abstract

We discuss a class of filiform Lie superalgebras L^n,m. From these Lie superalgebras, all the other filiform Lie superalgebras can be obtained by deformations. We have decompositions of \mathrm{D}\mathrm{e}{\mathrm{r}}_{\overline{\mathrm{0}}}(L^n,m) and \mathrm{D}\mathrm{e}{\mathrm{r}}_{\mathrm{1}} (L^n,m). By computing a maximal torus on each L^n,m, we show that L^n,m are completable nilpotent Lie superalgebras. We also view L^n,m as Lie algebras, prove that L^n,m are of maximal rank, and show that L^n,m are completable nilpotent Lie algebras. As an application of the results, we show a Heisenberg superalgebra is a completable nilpotent Lie superalgebra.

Key words： Filiform Lie superalgebra; Heisenberg superalgebra; completable nilpotent Lie superalgebra; maximal torus; complete Lie superalgebra

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