Modules Of 2-toroidal Algebras And Kirillov-Reshetikhin Modules

This thesis is mainly to study the modules for 2-toroidal Lie algebras. Based on the triangular decomposition and PBW theorem, we construct the highest weight modules,and then study their properties including the irreducibility and integrability. Meanwhile,with the base of loop algebras, we study the fusion product of Kirillov-Reshetikhin modules and the various graded characters.In Chapter 1, we describe the research background and significance of this thesis,and briefly introduce the history and development of infinite dimensional Lie algebras.We discuss the representation and classification of modules for toroidal Lie algebras.Moreover, we introduce the definition of Kirillov-Reshetikhin modules and their graded characters.In Chapter 2, we present the basic definition and important conclusions of finite and infinite dimensional Lie algebras. It is the fundamental part of this thesis. The key point is to propose the toroidal Lie algebras, which is one of the classical examples of infinite Lie algebras.In Chapter 3, imaginary Verma modules, parabolic imaginary Verma modules, and Verma modules at level zero for 2-toroidal Lie algebras are constructed using three different triangular decompositions. Their relations are investigated, and several results are generalized from the affine Lie algebras. In particular, imaginary highest weight modules,integrable modules, and irreducibility criterion are also studied.In Chapter 4, highest weight modules of the 2-toroidal Lie algebra sl2 are studied under a new triangular decomposition. Singular vectors of Verma modules are determined using a similar condition with horizontal affine Lie subalgebras, and highest weight modules are described under the condition that c1> 0 and c2= 0.In Chapter 5, we introduce the Kirillov-Reshetikhin modules including their tensor product and fusion product. In particular, we study their graded multiplicities and graded characters. For slr+1, Francesco and Kedem had constructed the generalized Macdonald difference operators which satisfy the dual quantum Q system, then got the explicit expressions of graded characters. By the slight changes of the dual quantum Q system,we analysis the properties of the graded characters. In addition, we present another matrix representation of the graded characters.