量子群及相关代数的表示理论

本项目主要研究与李代数、量子群密切相关的一些结合代数的表示理论及其交叉，具体包括：.1、研究仿射及分圆Birman-Murakami-Wenzl代数的范畴版本及表示论；.2、研究B,C,D型量子群与分圆Birman-Murakami-Wenzl代数之间的Schur-Weyl对偶；.3、研究与分圆Birman-Murakami-Wenzl代数、分圆Nazarov-Wenzl代数有关的范畴化。.本项目研究的是李理论中基本且经典的对象，相关问题是表示论中的基本问题。其研究成果将极大促进李代数、量子群的表示理论的发展和交叉。

This is a proposal on the representation theory of some associative algebras, which are closely related to Lie algebras and quantum groups. We are mainly interested in the following problems. The first one is about the representations of the category version of affine and cyclotomic Birman-Murakami-Wenzl algebras. The second one is about the Schur-Weyl duality between quantum groups in types B、C、D and cyclotomic Birman-Murakami-Wenzl algebras. The third one is about the categorifications related to cyclotomic Birman-Murakami-Wenzl algebras and cyclotomic Nazarov-Wenzl algebras. The proposal addresses the classical and fundamental objects and problems in representation theory. The research will lead to the important development of the representation theory.