论文标题
代表理论中关于双重性ABC的讲座
Lectures on dualities ABC in representation theory
论文作者
论文摘要
在这些夏季迷你课程的讲义中，我们提供了量子组和Hecke代数的博览会，包括（准）R-Matrix，规范基础和$ q $ -schur二元性。然后，我们在由量子对称对引起的$ \ imath $量子群的设置中制定对应物，包括（quasi）k-matrix，$ \ imath $ car-caronical基础和$ \ imath $ schur duality。作为一个应用程序，（$ \ imath $ - ）规范基础用于制定lie（超级）A-D型代数的BGG类别中的Kazhdan-lusztig理论和字符公式。最后，提供了$ q $ -schur和$ \ imath $ schur二元的几何构造。
In these lecture notes for a summer mini-course, we provide an exposition on quantum groups and Hecke algebras, including (quasi) R-matrix, canonical basis, and $q$-Schur duality. Then we formulate their counterparts in the setting of $\imath$quantum groups arising from quantum symmetric pairs, including (quasi) K-matrix, $\imath$-canonical basis, and $\imath$Schur duality. As an application, the ($\imath$-)canonical bases are used to formulate Kazhdan-Lusztig theories and character formulas in the BGG categories for Lie (super)algebras of type A-D. Finally, geometric constructions for $q$-Schur and $\imath$Schur dualities are provided.
