Maximal Subalgebras Of Lie(Super)Algebras Of Cartan Type And Representations Of Filiform Lie Superalgebras

Lie(super)algebras are an important class of non-associative algebras, which are closed realted to many branches of mathematics. The theory of Lie(super)algebras is an efficient tool for analyzing the properties of physical systems. In mathematics, a Lie superalgebra is a generalisation of a Lie algebra, and a Lie algebra is also a Lie superalgebra. According to the underlying field, Lie superalgebras can be divided into two parts: modular Lie superalgebras(Lie superalgebras over a field of characteristic p > 0)and non-modular Lie superalgebras(Lie superalgebras over a field of characteristic zero).Lie superalgebras can be divided into simple Lie superalgebras and non-simple Lie superalgebras. For modular Lie(super)algebras, the main research are the structure and representations of simple modular Lie(super)algebras. Lie(super)algebras of Cartan type are important simple modular Lie(super)algebras. For simple Lie superalgebras over a field of characteristic zero, there are much beautiful research. More and more literatures focus on the non-simple Lie superalgebras, in particular, for nilpotent and solvable Lie superalgebras. Filiform Lie superalgebras are important nilpotent Lie superalgebras. This thesis is devoted to studying maximal graded subalgebras of restricted Lie superalgebras of Cartan type, maximal graded subalgebras of non-restricted Lie algebras of Cartan type and representations of filiform Lie superalgebras.Firstly, we determine maximal graded subalgebras of the four infinite series of simple modular Lie superalgebras over a field of characteristic p > 3. We give necessary and sufficient conditions for two maximal graded subalgebras to be conjugate, and then give a classification of maximal graded subalgebras of these Lie superalgebras under conjugation. We also give the conjugate classes and dimension formulas for all maximal graded subalgebras except for irreducible maximal graded subalgebras. These four series form a remarkable class of simple finite-dimensional Lie superalgebras, called the Lie superalgebras of odd Cartan type, since they occur(among finite-dimensional algebras) neither in the “non-super” case, nor in the case of characteristic zero. Thus, the classification of maximal graded subalgebras of these algebras is meaningful. Since the Lie superalgebras of odd Cartan type are generated by their “local part”, the classification follows by a“degree reduction” technique. The structure of 1-gradation as a module of 0-gradation is described. Then all maximal graded subalgebras are completely determined by virtue of a constructive method and the method of weight space decompositions.Then, we determine maximal graded subalgebras of non-restricted simple modular Lie algebras of Cartan type over a field of characteristic p > 5. All maximal graded subalgebras are completely determined except for irreducible maximal graded subalgebras.The classification of irreducible maximal graded subalgebras is reduced to that of the irreducible maximal subalgebras of the classical Lie algebras. The restricted Lie algebras of Cartan type are closed related to the infinite dimensional Lie algebras over a filed of characteristic zero. There are much beautiful research for maximal graded subalgebras of restricted Lie algebras of Cartan type. The sturcture of non-restricted Lie algbras of Cartan type rely heavily on the underlying field. The classification of maximal graded subalgebras is more complicated. Motivated by the work of the classification of maximal graded subalgebras for Lie superalgebras of odd Cartan type, we introduce quasi-maximal graded subalgebras for the first time. We show that all maximal graded subalgebras are quasi-maximal graded subalgebras; and then give necessary and sufficient conditions for quasi-maximal graded subalgebras to be maximal. Since non-restricted modular Lie algebras of Cartan type can not generated by their local part, we find the generators of these Lie algebras, and then determine the maximal graded subalgebras for these Lie algebras by virtue of a constructive method and the method of weight space decompositions.At last, we study the representations of model filiform Lie superalgebras. We determine the minimal dimension of a faithful module over the model filiform Lie superalgebras and; construct some finite or infinite dimensional modules over the model filiform Lie superalgebras on the Grassmann algebra and the mixed Clifford-Weyl algebra. Filiform Lie superalgebras are a particular class of nilpotent Lie superalgebras. All filiform Lie superalgebras can be obtained by infinitesimal deformations of the model filifrom Lie superalgebras. We mainly use the constructive method and the properties of Jordan canonical forms to determine the minimal dimensions of faithful modules for the model filiform Lie superalgebras. Moreover, we get lots of representations for the model filiform Lie superalgebras by considering representations over general linear Lie superalgebras such as the representations constructed by Feingold A and Frenkel.