Derivations for the even parts of modular Lie superalgebras $W$ and $S$ of Cartan type

Abstract

Let $\mathbb{F}$ be the underlying base field of characteristic $p>3$ and denote by $\mathcal{W}$ and $\mathcal{S}$ the even parts of the finite-dimensional generalized Witt Lie superalgebra $W$ and the special Lie superalgebra $S,$ respectively. We first give the generator sets of the Lie algebras $\mathcal{W}$ and $\mathcal{S}.$ Using certain properties of the canonical tori of $\mathcal{W}$ and $\mathcal{S},$ we then determine the derivation algebra of $\mathcal{W}$ and the derivation space of $\mathcal{S}$ to $\mathcal{W},$ where $\mathcal{W}$ is viewed as $\mathcal{S}$-module by means of the adjoint representation. As a result, we describe explicitly the derivation algebra of $\mathcal{S}.$ Furthermore, we prove that the outer derivation algebras of $\mathcal{W}$ and $\mathcal{S}$ are abelian Lie algebras or metabelian Lie algebras with explicit structure. In particular, we give the dimension formulae of the derivation algebras and outer derivation algebras of $\mathcal{W}$ and $\mathcal{S}.$ Thus we may make a comparison between the even parts of the (outer) superderivation algebras of $W$ and $S$ and the (outer) derivation algebras of the even parts of $W$ and $S,$ respectively.

52 pages

Cited by in corpus (2)

• Derivations from the even parts into the odd parts for Lie superalgebras W and S
• Associative forms and second cohomologies of Lie superalgebras $HO$ and $KO$