Center Of The Quantum Weyl Symmetric Polynomials And Quantum Groups

Quantum group or Drinfeld-Jimbo quantized enveloping algebra developing in the mid of the eighties is one of the important branches of algebra. Since quantum group has the fundamental algebra structure behind mathematical physics, and the tight relation with other branches of mathematics, it has developed rapidly during the last twenty years. The aim of this dissertation is to describe explicitly the generators and generating relation for the center of U q ( sl n)( while n = 3 and 4) by Harish-Chandra isomorphism and properties of the quantumWeyl elementary symmetric polynomials, where U q ( sl n)is the quantized enveloping algebra of sln .In the first part, we introduce the main results of quantum group U q( g ). Meanwhile, we also brief the main methods and some important results obtained in this dissertation.In the second part, we recall some basic concepts of Lie algebra and its root system, Cartan matrix, together with quantized enveloping algebra. Moreover, we mainly introduce some basic results of U q( g ), such as: the graded structure of U q( g ),triangular decomposition , and the Harish-Chandra isomorphismIn the third part, we first review the root system, root base, weight lattice, and fundamental weight of sl3 . We also recall the relations between root base and correspon- ding to the fundamental weight, and the action ofWeyl group on its root system. Next, by computing the relations between root lattice and"even"weight lattice, and using the action ofWeyl group on Uq (s l3 )0e v, we proveNow, for any t , s∈, we call is the third quantumWeyl symmetric polynomials. In particular, let We call x₁ , x₂ ,x₃ are the third quantumWeyl elementary symmetric polynomials. It turns out that x₁ , x₂ ,x₃ are minimal generators of (U q (s l3 )0e v)W and the minimal generators x₁ , x₂ ,x₃ are satisfied the expression(6). Then, by Harish-Chandra isomorphism, we obtain one of the main results in this paper, as follows:Theorem 3.2.6 The center Z (U_q( sl₃ ))of U_q( sl₃ )is isomorphic to k [ y₁ , y₂ , y₃ ]/ I , where I is the ideal generated byFrom this theorem, we describe explicitly the structure of Z (U_q( sl₃ )).In the fourth part, we first recall the root system, root base, weight lattice, and fundamental weight of sl4 .We also review the relations between root base and correspon- ding to the fundamental weight, and the action ofWeyl group on its root system. Next, by computing the relations between root lattice and"even"weight lattice, and using the action ofWeyl group on U_q(sl₄ )0ev we proveNow, for any t , s , m∈, we call is the fourth quantumWeyl symmetric polynomials. In particular, letWe call x₁ , x₂ , x₃ ,x₄ are the fourth quantumWeyl elementary symmetric polynomials. It turns out x₁ , x₂ , x₃ ,x₄ are minimal generators of (U_q (sl₄ )e0 v)W and the minimal generators x₁ , x₂ , x₃ ,x₄ are satisfied the expression(11). Then, by Harish-Chandra isomorphism, we obtain another main result in this paper, as follows:Theorem 4.2.6 The center Z (U_q( sl₄))of U_q( sl₄)is isomorphic to k [ y₁ , y₂ , y₃ , y₄]/ I , where I is the ideal generated byFrom this theorem, we describe explicitly the structure of Z (U q( sl4 )).