DERIVATIONS OF THE EVEN PARTS FOR MODULAR LIE SUPERALGEBRAS OF CARTAN TYPE W AND S

Abstract

Let 𝔽 be the underlying base field of characteristic p < 3 and denote by [Formula: see text] and [Formula: see text] 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 [Formula: see text] and [Formula: see text]. Using certain properties of the canonical tori of [Formula: see text] and [Formula: see text], we then determine the derivation algebra of [Formula: see text] and the derivation space of [Formula: see text] to [Formula: see text], where [Formula: see text] is viewed as a [Formula: see text]-module by means of the adjoint representation. As a result, we describe explicitly the derivation algebra of [Formula: see text]. Furthermore, we prove that the outer derivation algebras of [Formula: see text] and [Formula: see text] are abelian Lie algebras or metabelian Lie algebras with explicit structure. In particular, we give the dimension formulas of the derivation algebras and outer derivation algebras of [Formula: see text] and [Formula: see text]. 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.