On bases of quantum affine algebras

TL;DR

This paper explores algebraic methods for constructing the canonical basis of the positive part of quantum affine algebras, extending finite type techniques to affine types, with key results from several researchers.

Contribution

It provides algebraic constructions of the affine canonical basis, building on and generalizing finite type methods to the affine case.

Findings

Algebraic constructions for affine canonical basis are established.

Results from Beck-Nakajima, Lin-Xiao-Zhang, Xiao-Xu-Zhao are summarized.

Extension of finite type methods to affine type is achieved.

Abstract

Let $\textbf{U}^+$ be the positive part of the quantum group $\textbf{U}$ associated with a generalized Cartan matrix. In the case of finite type, Lusztig constructed the canonical basis $\textbf{B}$ of $\textbf{U}^+$ via two approaches. The first one is an elementary algebraic construction via Ringel-Hall algebra realization of $\textbf{U}^+$ 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.