Rational Ringel-Hall algebras, Hall polynomials of affine type and Canonical bases

TL;DR

This paper introduces rational Ringel-Hall algebras for tame quivers, establishes Hall polynomial existence, constructs PBW bases, and develops canonical bases for quantum extended Kac-Moody algebras, advancing understanding of their algebraic structure.

Contribution

It defines rational Ringel-Hall algebras for tame quivers and constructs canonical bases, linking Hall polynomials with quantum algebra structures.

Findings

Existence of Hall polynomials for tame quiver algebras

Construction of PBW bases for these algebras

Development of canonical bases with strong purity properties

Abstract

In this paper, the rational Ringel-Hall algebras for tame quivers are introduced and are identified with the positive part of the quantum extended Kac-Moody algebras. By using the rational Ringel-Hall algebras, we show that the existence of Hall polynomials for tame quiver algebras. The PBW bases are constructed and new classes of perverse sheaves are shown to have strong purity property. These allows us to construct the canonical bases of the positive part of the quantum extended Kac-Moody algebras.