Drinfel'd Realizations, Quantum Affine Lyndon Bases And Vertex Representations Of Two-parameter Quantum Affine Algebras

In recent years, more and more people are interested in studying the two-parameter quantum groups. In 2001, motivated by down-up algebras, Benkart-Witherspoon introduced the structure of the two-parameter quantum groups of type of gl_n, sl_n, and investigated their representation theory and center as well as R-matrix ([5]-[7]). In 2004, Bergeron-Gao-Hu ([9], [10]) extended it to the structures of two-parameter quantum groups of type of finite B, C, D. At the same time, they investigated the representation theory and the environment condition upon Lusztig's symmetries existence of these quantum groups. Later on, Hu-Shi and Bai-Hu studied the two-parameter quantum group of type G₂ and E₆ in [35] and [2], respectively. Motivated by these works on two-parameter quantum groups corresponding to finite-dimensional simple Lie algebras, it is natural to establish the theory of two-parameter quantum affine algebras. With continuing the work on U_r,s(A_n-2⁽¹⁾, in this thesis, we present the structures of two-parameter quantum groups corresponding to all non-twisted affine Lie algebras, show that these quantum affine algebras can be realized as Drinfel'd doubles of certain Hopf subalgebras with respect to skew-dual pairing, and derive the Drinfel'd realizations for two-parameter cases. On the one hand, depending completely on our adopted combinatorial approach, we obtain the quantum affine Lyndon bases of two-parameter quantum affine algebras. On the other hand, we state and prove the isomorphism theorem for Drinfel'd realizations of two-parameter quantum affine algebras in detail. Finally, we construct the level one vertex representations of two-parameter quantum affine algebras for all non-twisted cases.The first main content of this thesis is to define the structures of two-parameter quantum groups corresponding to all non-twisted affine Lie algebras. In other words, the defining system is a set of defining generators and defining relations in the sense of Benkart-Witherspoon. As Hopf algebras, we indicate the Drinfel'd doubles properties of two-parameter quantum affine algebras.In order to explore further and enrich the structure and representation theory of the two-parameter quantum affine algebras later on, the second main aim of this thesis isto work out the Drinfel'd realizations of two-parameter quantum affine algebras. To do it, from the antisymmetric point of view, we have used the Q-algebra antiautomorphism Ï„. At the same time, our method, to some extent, follows the approach to a kind of description of the quantum affine Lyndon bases. Actually, we can construct explicitly all quantum real and imaginary root vectors as well as quantum affine Lyndon bases using our method.As well-known, in 1987, Drinfel'd gave an important conjecture about the realizations of classical quantum affine algebras, but he didn't prove it at that time. Afterwards, The proof for some non-twisted cases wase studied by many authors, for instance, Damiani([15]), Beck ([3]), Jing ([42]) etc. The third main content of this thesis is to give the non-twisted Drinfel'd isomorphism for our two-parameter cases and prove it case by case. The proof depends completely on our adopted combinatorial approach with specific techniques. If the readers go with us into the details, they will find how our technical calculations (in somehow a bit tedious) work well and necessarily for exactly verifying the compatibilities of the defining system.Based on the rich works on the vertex representations theory of classical quantum affine algebras ([27], [11], [41], [43]-[53] etc.), we construct the level-one vertex representations for all non-twisted two-parameter quantum affine algebras which is the last main content of this thesis. This chapter consists of four parts. At first, we introduce the vertex operators of simply-laced cases and construct their level-one vertex representations. Next, we obtain three level-one irreducible fundamental modules of two-parameter quantum affine algebra corresponding to affine Lie algebra of type B. In the case of two-parameter quantum affine algebras U_r,s(C_n⁽¹⁾) or U_r,s(F₄⁽¹⁾), the level-one vertex modules we construct are reducible owing to the existence of bosonic operators. Finally, we establish the level-one vertex representation of two-parameter quantum affine algebras...