Derivations of the even part of finite-dimensional simple modular Lie superalgebra $$\mathcal{M}$$

Abstract

Let \(\mathbb{F}\) be the underlying base field of characteristic p > 3 and denote by \(\mathfrak{M}\) the even part of the finite-dimensional simple modular Lie superalgebra \(\mathcal{M}\). In this paper, the generator sets of the Lie algebra \(\mathfrak{M}\) which will be heavily used to consider the derivation algebra \(Der\left( \mathfrak{M} \right)\) are given. Furthermore, the derivation algebra of \(\mathfrak{M}\) is determined by reducing derivations and a torus of \(\mathfrak{M}\), i.e., $$Der(\mathfrak{M}) = ad(\mathfrak{M}) \oplus span_\mathbb{F} \left\{ {\prod\limits_l {ad(\xi _{r + 1} \xi _l )} } \right\} \oplus span_\mathbb{F} \left\{ {adx_{i'} ad\left( {x_i \xi ^v } \right)\prod\limits_l {ad(\xi _{r + 1} \xi _l )} } \right\}.$$ . As a result, the derivation algebra of the even part of M does not equal the even part of the derivation superalgebra of M.