Completable nilpotent Lie superalgebras

We discuss a class of filiform Lie superalgebras Ln,m. From these Lie superalgebras, all the other filiform Lie superalgebras can be obtained by deformations. We have decompositions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Der_{\bar 0} \left( {L^{n,m} } \right)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Der_{\bar 1} \left( {L^{n,m} } \right)$$\end{document}. By computing a maximal torus on each Ln,m, we show that Ln,m are completable nilpotent Lie superalgebras. We also view Ln,m as Lie algebras, prove that Ln,m are of maximal rank, and show that Ln,m are completable nilpotent Lie algebras. As an application of the results, we show a Heisenberg superalgebra is a completable nilpotent Lie superalgebra.