Simple Modules For Some Cartan Type Lie Superalgebras

In this paper, the Cartan type Lie superalgebras over an algebraic closed fieldof positive characteristic are considered.First, we recall the definitions of Lie superalgbras W (n), S(n), K(n), W,S, H, K, HO, KO. We construct a series of modules ML(S(λ)), induced fromthe restricted universal enveloping superalgebras of Lie superalgebras W (n), S(n),K(n), W, S, H, K, HO, KO. With some special weights λ, we obtain the sufcientconditions that ML(S(λ)) are irreducible L-modules, where L=W (n), S(n), K(n),W, S, H, K, HO, KO.Then, we recall the definition of a new class of finite-dimensional simple modu-lar Lie superalgebras Ω. Prof. Zhang Yongzheng constructed the finite-dimensionalsimple modular Lie superalgebras Ω and determined its derivation superalgbera in2009. Since, the superderivations algebra of a Lie superalgebra is still a Lie super-algebra, it’s nature for us to consider whether the even part of the superderiva-tion algebra of Lie superalgebra equals the derivation algebra of the even part ofLie superalgebra or not. We first give the generator sets of the Lie algebra Ω.Then, we reduce the derivation of Ω to a certain form. With the reduced deriva-tion and a torus of, we determine the derivation algebra of Ω. We obtainedDer(Ω0)=Der(Ω)0.