Representations Of Cartan Type Modular Lie Superalgebras And Abelian Lie Superalgebras

As an important part of superalgebra category,Lie superalgebras not only are closely related to many branches of mathematics,but also have classical physical background.According to the characteristic of 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).Representation plays an important role in the algebraic theory.The present thesis is devoted to studying representations of Cartan type modular Lie superalgebras and the minimal faithful representations of abelian Lie superalgebras.Firstly,we study the simplicity of restricted Kac modules by use of root reflections and induced modules: by use of the fact that any restricted Kac module has the unique maximal submodule,all the restricted simple modules of the superalgebras under consideration are reduced to simple heads of restricted Kac modules,the restricted Kac modules with respect to the standard Borel subalgebra are reduced to restricted Kac modules with respect to a series of Borel subalgebras,and the relations of the corresponding simple heads are reduced to the ones of the highest weights.In particular,we give a sufficient and necessary condition of restricted Kac modules to be simple in terms of atypical and typical weights.Secondly,we study the character formulas of restricted simple modules for Cartan type Lie superalgebras by use of the theory of rational modules.By use of isomorphisms of simple heads for restricted Kac modules and the forms of maximal vectors,the composition factors and the corresponding short exact series of restricted Kac modules are obtained.In particular,the character formulas and dimensions of all the restricted simple modules for the superalgebras under consideration are obtained.Thirdly,we study the non-restricted representations of Cartan type Lie superalgebrs by use of the height of p-characters and the PBW theorem of χ-reduced Kac modules.The non-restricted simple modules of superalgebras under consideration are reduced to the simple heads of χ-reduced Kac modules,and the study can be reduced to ranks for matrices of p-charater.In particular,we characterize all simple g-modules with nonsingular or ?-invertible p-characters.We also obtain all simple g-modules with regular semisimple p-characters.Fourthly,we study the minimal dimensions of faithful representations for finite dimensional abelian Lie superalgebras over a field of characteristic zero by use of the similar transformation of matrices.We introduce a new version of general linear Lie superalgebra and study the maximal dimensions of abelian subalgebras contained in upper triangular matrices algebras.In particular,we also obtain the maximal dimensions of abelian subalgebras for the general linear Lie superalgebras and the minimal dimensions of faithful representations for abelian Lie superalgebras.Lastly,we characterize the maximal abelian subalgebras by use of the new version isomorphic to the general linear Lie superalgebra.By use of the two operators,we obtain a sufficient and necessary condition for the dimensions of abelian subalgebras to be maximal of the general linear Lie superalgebra.Then all the maximal abelian subalgebras of the general linear Lie superalgebra can be classified in the sense of conjugation and all the minimal faithful representations of Nice abelian Lie superalgebras.