Tame quivers and affine enveloping algebras

TL;DR

This paper constructs integral bases for the positive part of affine Kac-Moody algebras' enveloping algebra using Hall algebra techniques, leveraging tame quiver representation theory.

Contribution

It introduces a novel Hall algebra approach to explicitly construct integral bases for the enveloping algebra of affine Kac-Moody algebras, connecting quiver representations with algebraic structures.

Findings

Explicit integral bases for $U( ^+)$ constructed

Hall algebra approach effectively applied to affine Kac-Moody algebras

Representation theory of tame quivers is central to the construction

Abstract

Let $\mathfrak{g}$ be an affine Kac-Moody algebra with symmetric Cartan datum, $\mathfrak{n^{+}}$ be the maximal nilpotent subalgebra of $\mathfrak{g}$. By the Hall algebra approach, we construct integral bases of the $\mathbb{Z}$-form of the enveloping algebra $U(\mathfrak{n^{+}})$. In particular, the representation theory of tame quivers is essentially used in this paper.