Mathematical Sciences: Bialgebra Structures in the Theory ofRepresentations of Quantum Groups and the Quantum Yang-Baxter Equations

本项目涉及包络代数的量子类似物的构造，它们的表示理论，以及在表示空间上的量子Yang-Baxter方程的解。首席研究员将使用模块理论方法来求解量子Yang-Baxter方程。特别强调的是在二维情况下对解进行分类。有限维拟三角形Hopf代数也将被研究，并将重点放在范畴构造上。量子群对于数学家和物理学家来说都是一个新的研究领域。在数学方面，它结合了“纯粹”数学的三个最古老的领域：代数、分析和几何，但它对研究共形量子场论的物理学家非常感兴趣。

This project is concerned with the construction of the quantum analogs of enveloping algebras, their representation theory, and solutions to the quantum Yang-Baxter equation on representation spaces. The principal investigator will use a module theoretic approach to the quantum Yang-Baxter equation. Particular emphasis will be given to classifying solutions in the 2-dimensional case. The class of finite dimensional quasitriangular Hopf algebras will also be studied with attention given to categorical constructions. Quantum groups are a new area of research for both mathematicians and physicists. On the mathematical side, it combines three of the oldest areas of "pure" mathematics, algebra, analysis and geometry, yet it is of great interest to physicists working in conformal quantum field theory.