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