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 $$Der_{\bar 0} \left( {L^{n,m} } \right)$$ and $$Der_{\bar 1} \left( {L^{n,m} } \right)$$ . 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.