Mathematical Sciences: Representation Theory: Quantum Groupsand Algebraic Groups

本文主要研究量子群的表示理论及其与代数群的模表示理论的关系。主要研究Weyl模和量子Weyl模的Jantzen滤子的相容性；在非普通位置具有最高权的Weyl模的结构；与代数群和量子群的诱导函子有关的上同调模的结构，以及它们之间的关系。群是在其上定义了运算的代数结构。如果一个群的元素满足一个多项式恒等式，则称它为代数群。代数群对于数学和物理学的几个不同领域都很重要。

This research is concerned with the representation theory of quantum groups and its relation with the modular representation theory of an algebraic group. The principal investigator will study the compatibility of the Jantzen filtrations for Weyl modules and quantum Weyl modules; the structure of Weyl modules with highest weights in the non-generic position; and the structure of cohomology modules related to induction functors for both the algebraic group and the quantum groups, and the relations between them. A group is an algebraic structure with an operation defined on it. A group is called an algebraic group if its elements satisfy a polynomial identity. Algebraic groups are important for several different areas of mathematics, as well as physics.