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.

Keywords

Modular Lie superalgebra / Derivation algebra / Generator sets

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