Wedge Modules For Two-parameter Quantum Groups

In this paper, we construct all the wedge modules for the two-parameter quantumgroup U_r,s（sl_n） by using the fusion rule. Although the tensor modules of U_r,s（sl_n） havebeen constructed by Benkart G. and Witherspoon S., it is nontrivial to pass down to theirreducible quotient modules. The fundamental representations of the quantum groupU_q（（sl_n）|∧） were first constructed by Rossor in [29]. Later the fundamental modules were re-constructed by using the fusion procedure in [25,23,21] in connection with the quantumafne algebras U_q（（sl_n）|∧）. We extend their work of one-parameter quantum group to the caseof two-parameter quantum group U_r,s（sl_n）. By using the fusion procedure, we can alsoconstruct all wedge modules for the two-parameter quantum group U_r,s（sl_n）. The result issimilar to the case of one-parameter. In order to carry out the fusion procedure, we needto find the R-matrix with spectral parameter. This idea is called Yang-Baxterization,which is from the paper [10] written by Ge M.L., Wu Y.S. and Xue K..The main structure of the thesis is as follows:In chapter I, we introduce the backgrounds and modern developments of the two-parameter quantum group U_r,s（sl_n） as well as the results of this paper.In chapter II, we first give a review of the fundamental concepts, such as Hopf algebra,two-parameter quantum groups, the highest weight modules and so on. Then we introducethe relations between the main diagonal and sub-diagonal elements of triangular matrixT^{（+）}, T^{（-）}and the generators of U_r,s（gl_n）, U_r,s（sl_n）. Finally, we introduce some results ofone-parameter in order to compare with the case of two-parameter.In chapter III, we construct all the wedge modules for the two-parameter quantumgroup by using the fusion procedure. This chapter is divided into three steps. Firstly,we use the Yang-Baxterization method to construct a spectral parameter dependent R-matrix R（z） for the two-parameter quantum algebra U_r,s（sl_n） based on the braid grouprepresentation given by the Benkart-Witherspoon R-matrix. Secondly, we construct the（r,s）-symmetric tensors S_r,s²ï¼ˆV） as well as the anti-symmetric tensors Λ_r,s²ï¼ˆV） by usingthe fusion procedure, which are as the same as the results of Benkart-Witherspoon in[3]. Thirdly, using the fusion procedure, we determine all （r,s）-wedge modules for thetwo-parameter quantum group U_r,s（sl_n）, which is isomorphic to the corresponding funda-mental repersentation V （（ωk）|-）. In each step, we do the comparation between the case of two-parameter and the case of one-parameter.