Canonical bases for the quantum extended Kac-Moody algebras and Hall polynomials

TL;DR

This paper constructs canonical bases for quantum extended Kac-Moody algebras using Hall algebra techniques, establishing new connections between algebraic and geometric methods, and proves the existence of Hall polynomials for tame quivers.

Contribution

It introduces a new approach to canonical bases via Hall algebras and proves Hall polynomial existence for tame quivers, advancing the understanding of quantum Kac-Moody algebras.

Findings

Constructed PBW basis for the algebra

Established isomorphism with the singular Ringel-Hall algebra

Proved the existence of Hall polynomials for tame quivers

Abstract

In this paper, the singular Ringel-Hall algebra for a tame quiver is introduced and shown to be isomorphic to the positive part of the quantum extended Kac-Moody algebra. A PBW basis is constructed and a new class of perverse sheaves is shown to have purity property. This allows to construct the canonical bases of the positive part of the quantum extended Kac-Moody algebra. As an application, the existence of Hall polynomials for tame quiver algebras is proved.