A Study Of Correlative Problems For Lie Algebras And (?)-Graded Lie Superalgebras

The study of Lie algebras has three major directions:finite dimensional Lie algebras over fields of characteristic zero, modular Lie algebras(that is, Lie algebras over fields of prime characteristic), Kac-Moody Lie algebras and other infinite dimensional Lie algebras. As is well known, the research of finite dimensional Lie algebras and infinite dimensional Lie algebras of characteristic zero has obtained remarkable evolutions. For instance, the classifications of general finite dimensional simple Lie algebras and finite dimensional restricted simple Lie algebras, and the theories of Kac-Moody Lie algebras, Virasoro Lie algebras, Witt Lie algebras have been settled. However, many problems are still open for infinite dimensional Lie algebras, for instance, the classification for infinite dimensional simple Lie algebras and the research for affine modular Lie algebras have not been settled. Derivation algebras are important tools to study Lie algebras, in recent years, many researchers have investigated derivation algebras of matrix algebras and generalized the definition of derivations and obtained more Lie algebras containing derivation algebras. We also know that derivation algebras of Lie algebras are closely related to the cohomology groups, moreover, the cohomology groups are related to the central extensions. The remarkable results for the cohomology groups and extensions of Lie algebras motivate researchers to study the same theories for Lie superalgebras.The purpose of this dissertation is to study Lie triple derivations and generalized Lie triple derivations of certain Lie algebras over the field of complex numbers and matrix algebras over commutative rings, an affine modular Lie algebra and its representation, and skew superderivations and P-associative forms of Z-graded Lie superalgebras.The first chapter contains some basic facts and motivations for this dissertation. In particular, we introduce some recent important developments in the related subjects.In Chapter 2, using root system, we study generalized Lie triple derivations for the Borel subalgebras b of complex semisimple Lie algebras and the maximal nilpotent subalgebras n of classical complex simple Lie algebras. For Lie triple derivations of n, we construct some standard Lie triple derivations and show the decomposition of any Lie triple derivation. What is more, the codimension of derivation algebras in the Lie triple derivation algebras and the solvability of Lie triple derivation algebras are characterized.In Chapter 3, following the work of Benkoric and D.Y.Wang, we study Lie triple deriva- tions from the Lie algebra consisting of all upper triangular matrices over a commutative ring to its 2-torsion free bimodule and Lie triple derivations for the parabolic subalgebras of general linear Lie algebra.In Chapter 4, applying knowledge of restricted Lie algebras, we study the affine Lie algebra L corresponding to special linear Lie algebra L= sl(l+1,K). We prove that L is a restricted Lie algebra and show some results for its restricted modules.In Chapter 5, we study skew superderivations and P-associative forms of Z-graded Lie superalgebras. The relation between skew superderivations and the bilinear forms associative with a subalgebra P is characterized.