Derivations Of The Even Parts For Generalized Witt Lie Superalgebras In Characteristic3

The studies of modular Lie superalgebras have obtained many important re-sults, but a lot of problems still have been unresolved. The even parts of Liesuperalgebras are Lie algebras, the odd parts are the natural modular of the evenparts. Therefore, it is significant to study the structure of the even parts of modularLie superalgebras.LetFbe an algebraically closed field of characteristic3and denote by W theeven parts of the finite-dimensional generalized Witt Lie superalgebras. In thispaper, we firstly find the so-called canonical torus of W and construct aZ-gradedsubalgebra M of W. Then we give the weight space decomposition of W withrespect to canonical torus of W and calculate the weight vectors in M with thesame weights for generators of W, then we determine the derivation algebra of W.In this paper, the derivations of W are completely determined by means of theweight space decomposition with respect to the canonical torus and constructing asuitableZ-graded subalgebra of W. This method can greatly simplify computationalprocess. In specified computational process, we firstly study the influence of thetop over the derivation of W. Then we calculate the derivations of nonnegativeZ-degree and negativeZ-degree. Finally we determine the derivation algebra of W.The method and results in this paper can be used to study derivations of the evenparts for other Lie superalgebras.