Derivations of the Odd Contact Lie Algebras in Prime Characteristic

Abstract

Abstract. The underlying field is of characteristic p > 2. In this paper, we first use the method of computing the homogeneous derivations to determine the first cohomology of the so-called odd contact Lie algebra with coefficients in the even part of the generalized Witt Lie superalgebra. In particular, we give a generating set for the Lie algebra under consideration. Finally, as an application, the derivation algebra and outer derivation algebra of the Lie algebra are completely determined. IntroductionIn this paper, we consider a family of non-simple modular Lie algebras, called the odd contact Lie algebras, which are actually the even parts of the odd contact simple Lie superalgebras that are also closely related to the contact simple Lie algebras. The main purpose is to compute the first cohomology of the odd contact Lie algebras with coefficients in their adjoint representations. In other words, we determine the derivation algebras and the outer derivation algebras.As is well-known, the theory of modular Lie algebras has undergone a remarkable evolution. In the super-case, the theory of modular Lie superalgebras has also obtained many interesting results in the past decade. For example, one can find work on the classification of classical modular Lie superalgebras [1, 2] and on the structures and representations of modular Lie superalgebras of Cartan type [6,7,9,16,17,18]. Recently, one can also find work on the representations of the classical modular Lie superalgebras (see, for example, [14,15]). For the simple modular Lie algebras and simple modular Lie superalgebras of Cartan type, the (super)derivation algebras have been sufficiently studied (for example, see [3,4,8,9,10,11,12,13]). In [4,11], the superderivation algebras of Lie superalgebras of Cartan-type H, W , S, K, HO and KO were determined. For the derivations of the even parts of Lie superalgebras