Completable nilpotent Lie superalgebras

We discuss a class of filiform Lie superalgebras L-n,L-m. From these Lie superalgebras, all the other filiform Lie superalgebras can be obtained by deformations. We have decompositions of Der((0) over bar)(L-n,L-m) and Der((1) over bar)(L-n,L-m). By computing a maximal torus on each L-n,L-m, we show that L-n,L-m are completable nilpotent Lie superalgebras. We also view L-n,L-m as Lie algebras, prove that L-n,L-m are of maximal rank, and show that L-n,L-m are completable nilpotent Lie algebras. As an application of the results, we show a Heisenberg superalgebra is a completable nilpotent Lie superalgebra.