On bases of quantum affine algebras

Abstract

Let ${\text{U}}^{+}$ be the positive part of the quantum group $\text{U}$ associated with a generalized Cartan matrix. In the case of finite type, Lusztig constructed the canonical basis $\text{B}$ of ${\text{U}}^{+}$ via two approaches. The first one is an elementary algebraic construction via Ringel-Hall algebra realization of ${\text{U}}^{+}$ and the second one is a geometric construction. The geometric construction of canonical basis can be generalized to the cases of all types. The generalization of the elementary algebraic construction to affine type is an important problem. We give several main results of algebraic constructions to the affine canonical basis in this ariticle. These results are given by Beck-Nakajima, Lin-Xiao-Zhang, Xiao-Xu-Zhao, respectively.

This review is based on the report of Jie Xiao given in the conference "Forty Years of Algebraic Groups, Algebraic Geometry, and Representation Theory in China" in Jan. 5th, 2020

References in corpus (1)

• Notes on affine canonical and monomial bases