2 5 Ju l 2 00 5 Derivations for the even parts of modular Lie superalgebras W and S of Cartan type

Let F be the underlying base field of characteristic p > 3 and denote by W and 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 W and S. Using certain properties of the canonical tori of W and S, we then determine the derivation algebra of W and the derivation space of S to W , where W is viewed as S-module by means of the adjoint representation. As a result, we describe explicitly the derivation algebra of S. Furthermore, we prove that the outer derivation algebras of W and 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 W and 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.