Higher Level Vertex Representation Of Two-parameter Quantum Affine Algebra U_r,s(C_n^（1）)

The two-parameter quantum group is a kind of generation of the quantum group. In2001, Benkart-Witherspoon defined the structure of the two-parameter quantum groups of type of gln, sln and investigated their representation theory. In2004, Bergeron-Gao-Hu extended the structures to the types of finite BCD. At the same time, they investigat-ed the highest weight representation theory. Later on, Hu-Shi and Bai-Hu studied the two-parameter quantum group of type G2and E6respectively. As the works about the two-parameter quantum groups for finite types has been completed, the subject of two-parameter quantum group for affine types becomes more and more important. First of all, Hu-Ross-Zhang studied two-parameter quantum affine algebra for affine An(1) type, defined their Drinfel’d realizations. In their further works, they constructed the level one vertex representation of two-parameter quantum affine algebra for all non-twisted cases. Furthermore, Jing and Zhang studied the representation theory of two-parameter quantum affine algebra. This current paper aim to construct the level-1/2vertex representation of two-parameter quantum affine algebra for type Cn(1), which can be realized the generation of one-parameter case. The paper is organized as follows.:Chapter2mainly introduces the basic knowledge of Lie algebra associated with this paper. Including the basic concept of lie algebra, the classification of single lie algebra, affine lie algebra’s theory and the basic information of the two-parameter quantum affine algebra;In chapter3, we introduce basic knowledge of simple Lie algebra and affine Lie algebra of type C, recall the definitions of two-parameter quantum affine algebra Ur,s(Cn(1)), at the same time, we also reviewed its Drinfel’d realization.In chapter4, we construct the Fock space firstly, then introduce two important operators Yi±(z) and Zi±(z). Because of the complexity of the two-parameter cases, We are particularly define a new cocycle. Finally, we construct the level-1/2vertex representation of two parameter quantum affine algebra Ur,s(Cn(1)) and check the operators satisfy all Drinfel’d relations.