Tame quivers and affine bases I: a Hall algebra approach to the canonical bases

TL;DR

This paper constructs a new bar-invariant basis for quantum affine algebras using tame quiver representations, showing it coincides with Lusztig's canonical basis and establishing a concrete bijection between them.

Contribution

It introduces a novel basis construction via a PBW basis from tame quiver representations, linking it explicitly to Lusztig's canonical basis.

Findings

The new basis matches Lusztig's canonical basis.

A concrete bijection between the bases is established.

The basis elements are indexed by modules and symmetric group characters.

Abstract

For quantum group of affine type, Lusztig gave an explicit construction of the affine canonical basis by simple perverse sheaves. In this paper, we construct a bar-invariant basis by using a PBW basis arising from representations of the corresponding tame quiver. We prove that this bar-invariant basis coincides with Lusztig's canonical basis and obtain a concrete bijection between the elements in theses two bases. The index set of these bases is listed orderly by modules in preprojective, regular non-homogeneous, preinjective components and irreducible characters of symmetric groups. Our results are based on the work of Lin-Xiao-Zhang and closely related with the work of Beck-Nakajima. A crucial method in our construction is a generalization of that by Deng-Du-Xiao.