论文标题
量子组和杨巴克斯方程的讲座
Lectures on Quantum Groups and Yang-Baxter Equations
论文作者
论文摘要
量子群理论的原理是从将它们用于物理模型中对称性变形的可能性的角度审查的。详细讨论了量子群理论的R-Matrix方法，并将其视为经典谎言组的量化以及一些谎言超级组的基础。我们首先阐明非共同和非协商HOPF代数的基础。 Hecke和Birman-Murakami-Wenzl（BMW）R-Matrices和相关的量子矩阵代数都非常关注。提出了与量子组GL_Q（N），SO_Q（N），SP_Q（2N）和SuperGroups GL_Q（N | M），OSP_Q（N | 2M）以及其合理（Yangian）限制相关的三角解。还考虑了Yang-Baxter方程的特殊谎言代数和椭圆解的有理R型。概述了辫子组的基本概念及其有限维度的商（例如Hecke和BMW代数）。给出了Hecke和BMW代数的表示理论的草图（包括查找基于群体及其量子维度的方法）。简要讨论了量子群和杨巴克斯特方程理论的应用。
The principles of the theory of quantum groups are reviewed from the point of view of the possibility of their use for deformations of symmetries in physical models. The R-matrix approach to the theory of quantum groups is discussed in detail and is taken as the basis of the quantization of classical Lie groups, as well as some Lie supergroups. We start by laying out the foundations of non-commutative and non-cocommutative Hopf algebras. Much attention has been paid to Hecke and Birman-Murakami-Wenzl (BMW) R-matrices and related quantum matrix algebras. Trigonometric solutions of the Yang-Baxter equation associated with the quantum groups GL_q(N), SO_q(N), Sp_q(2n) and supergroups GL_q(N|M), Osp_q(N|2m), as well as their rational (Yangian) limits, are presented. Rational R-matrices for exceptional Lie algebras and elliptic solutions of the Yang-Baxter equation are also considered. The basic concepts of the group algebra of the braid group and its finite dimensional quotients (such as Hecke and BMW algebras) are outlined. A sketch of the representation theories of the Hecke and BMW algebras is given (including methods for finding idempotents and their quantum dimensions). Applications of the theory of quantum groups and Yang-Baxter equations in various areas of theoretical physics are briefly discussed.
