Local (Anti-)Superderivations on Nilpotent Lie Superalgebras

Abstract

In this paper, we study local (anti-)superderivations on finite-dimensional nilpotent Lie superalgebras. Firstly, we prove that every finite-dimensional 2-step nilpotent Lie superalgebra over a field $\mathbb{F}$ with $\operatorname{char}\mathbb{F}\neq2$ admits pure local (anti-)superderivations (namely, local (anti-)superderivations that are not (anti-)superderivations). Then for $n$-step nilpotent Lie superalgebras over arbitrary fields with n greater than 2, we provide a sufficient criterion to guarantee the existence of pure local (anti-)superderivations. Furthermore, we show that 3-step nilpotent Lie superalgebras admit pure localsuperderivations.