2020/12/14 by Li Luo, Weiqiang Wang, Luo, Li +1
Mathematics · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2012.07203
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 \imathquantum groups arising from quantum symmetric pairs, including (quasi) K-matrix, \imath-canonical basis, and \imathSchur 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 \imathSchur dualities are provided.