On ${\mathrm{I}\mathrm{D}}^{\ast }$-superderivations of Lie superalgebras

Let $L$ be a Lie superalgebra over a field of characteristic different from $2,3$ and write ${\mathrm{I}\mathrm{D}}^{\ast }(L)$ for the Lie superalgebra consisting of superderivations mapping $L$ to $L^2$ and the central elements to zero. In this paper we first give an upper bound for the superdimension of ${\mathrm{I}\mathrm{D}}^{\ast }(L)$ by means of linear vector space decompositions. Then we characterize the ${\mathrm{I}\mathrm{D}}^{\ast }$-superderivation superalgebras for the nilpotent Lie superalgebras of class 2 and the model filiform Lie superalgebras by methods of block matrices.