Some Structures And Representations Of Lie Superalgebras

There are four main parts in this thesis.The first part is primarily concerned with generalized reduced Verma modules over Z-graded modular Lie super algebras. Some properties of the generalized reduced Verma modules and the coinduced modules are obtained. Moreover, the invariant forms on the generalized reduced Verma modules are considered. In particular, we prove that the generalized reduced Verma module is isomorphic to the mixed product for modules of Z-graded modular Lie superalgebras of Cartan type.The second part is primarily concerned with the natural filtrations of the finite-dimensional modular Lie superalgebras W(n, m) and S(n, m). By using their ad-nilpotent elements, the natural filtrations which are invariant relative to the automorphisms of the Lie superalgebras are employed in order to characterize the Lie superalgebras themselves.The third part is primarily concerned with the zero and first cohomology groups of gl(0,2) with coefficients in the generalized Witt Lie superalgebra, where gl(0,2) is the subalgebra of the general linear Lie superalgebra. Firstly, the zero cohomology group of gl(0,2) with coefficients in the generalized Witt Lie superalgebra is considered. Then the derivations and inner derivations from gl(0,2) into each submodules of the generalized Witt Lie superalgebra are calculated. Finally, the first cohomology group of gl(0,2) with coefficients in the generalized Witt Lie superalgebra is completely determined by matrices.The fourth part is primarily concerned with (α,β,γ)-superderivations of finite di-mensional Lie superalgebras over the field of complex numbers. Some properties of (α,β,γ)-superderivations of Lie superalgebras are obtained. In particular, two examples for (α,β,γ)-superderivations of low dimensional non-simple Lie superalgebras are pre-sented. Then the super-spaces of (α,β,γ)-superderivations for simple Lie superalgebras are determined. Using certain complex parameters we generalize the concept of cohomol-ogy cocycles of Lie superalgebras. A special case for the generalization of1-cocycles is exactly (α,β,γ)-superderivations. Furthermore, two-dimensional twisted cocycles of the adjoint representation are investigated in detail.