本项目将研究有限维广义Cartan型模李超代数W(n,m)、S(n,m)与H(n,m)的一类称为(α,β,γ)-超导子的广义超导子。W(n,m)是一类包含在U的所有超导子集合中的广义Witt 型模李超代数, 其中U是n元Grassmann代数与m元截头多项式的张量积。S(n,m)是一类广义特殊型模李超代数而H(n,m)是一类广义Hamiltonian型模李超代数。它们都是W(n,m) 的子代数。首先将分别得到W(n,m)、S(n,m)与H(n,m)的型心, 拟型心与δ-超导子的结构。进而将决定W(n,m)、S(n,m)与H(n,m)的(α,β,γ)-超导子空间。其次将得出一种特殊的广义1-上循环(被称为1-扭曲上循环)与(α,β,γ)-超导子的关系。最后还将对W(n,m)、S(n,m)与H(n,m)关于其伴随表示的低维扭曲上循环进行详细的研究。

This project will be concerned with a class of generalized superderivations called (α,β,γ)-superderivations for finite-dimensional generalized modular Lie superalgebras W(n,m), S(n,m) and H(n,m) of Cartan type. W(n,m) is a class of generalized modular Lie superalgebras of Witt type contained in the set of all superderivations of U, where U is the tensor product by the Grassmann algebra in n variables and the truncated polynomial algebra in m variables. S(n,m) is a class of generalized modular Lie superalgebras of special type and H(n,m) is a class of generalized modular Lie superalgebras of Hamiltonian type. They are subalgebras of W(n,m). Firstly, structures of centroids, quasicentroids and δ-derivations for W(n,m), S(n,m) and H(n,m) will be obtained, respectively. Then the super-spaces of (α,β,γ)-superderivations for W(n,m), S(n,m) and H(n,m) will be determined. Secondly, relations between a special case of generalized 1-cocycles (called 1-twisted cocycles) and (α,β,γ)-superderivations will be got. Finally, low-dimensional twisted cocycles of the adjoint representation for W(n,m), S(n,m) and H(n,m) will be investigated in detail.