A Study Of Lie Color Algebra And Correlative Problem

The present thesis is devoted to studying Lie color algebras and correlative problems(that is,Lie color algebras,Lie color triple systems,finite-dimensional simple Lie superalgebra over fields of prime characteristic).As is well known,the theories of Lie superalgebras of characteristic zero have seen remarkable evolutions,for instance,the classifications have been settled for finite-dimensional simple Lie superalgebras and infinite-dimensional simple linearly compact Lie superalgebras over algebraically closed fields of characteristic zero,respectively. On the other hand,the theory of Lie color algebras and Lie color triple systems are closely related to,but different from,Lie superalgebras of characteristic zero.Therefore,the work on Lie color algebras and Lie color triple systems are of particular interest.In particu-lar,many problem are still open for Lie color algebras and Lie color triple systems;for instance,the representative and the cohomology of Lie color algebras of classifi-cal type and classification have not been settled.We also know that six families of finite-dimensional simple modular Lie superalgebras of Cartan type have been found.What is more,the classification problem is still open for finite-dimensional simple modular Lie superalgebras.In Chapter 2 we give a definition of the cohomology of Lie color algebra and discuss some properties of derivations of Lie color algebras. Meanwhile,we obtain the relationship between skew derivation space and central extension on some Lie color algebras.Using this relationship we can study structure and central extension of Lie color algebras.In Chapter 3 we first introduce the notions of quadratic Lie color algebra and its double extension.We study the structure of quadratic Lie color algebra and obtain the sufficiency condition for a quadratic Lie color algebra to be a double extension. In Chapter 4 we discuss the completeness of some Lie color algebras and the sufficiency and necessary for the holomorph of a centerless perfect Lie color algebra to be complete.In Chapter 5 we study Lie color triple system.Speaking precisely,we first introduce the notions of Lie color triple system and quadratic Lie color triple system as well as Lie color triple system with some examples.We discuss some properties of derivation of Lie color triple systems. We also consider problem of the uniqueness of the decomposition of Lie color triple system whose center is zero. We show that a color symmetric invariant bilinear form on a Lie color triple system can be uniquely extended to its standard imbedding Lie color algebra.In Chapter 6 we construct a new family of finite-dimensional simple modular Lie superalgebraΩand prove its simplicity. By giving the generator set and formulating the homogeneous superderivations,we determine completely the su-perderivation algebra.Finally, we point out that this algebra isn't isomorphic to any known modular Lie superalgebras of Cartan-type.