The double Ringel-Hall algebra on a hereditary abelian finitary length category

TL;DR

This paper explores the structure of the Ringel-Hall algebra associated with semi-stable coherent sheaves over weighted projective curves, linking it to generalized Kac-Moody algebras and classifying indecomposable objects.

Contribution

It introduces the double Ringel-Hall algebra for a hereditary abelian finitary length category of semi-stable sheaves and establishes its relation to generalized Kac-Moody Lie algebras.

Findings

Defined the Ringel-Hall algebra of semi-stable sheaves

Connected the algebra to generalized Kac-Moody Lie algebras

Provided a Kac-type theorem for indecomposable semi-stable sheaves

Abstract

In this paper, we study the category $\mathscr{H}^{(\rho)}$ of semi-stable coherent sheaves of a fixed slope $\rho$ over a weighted projective curve. This category has nice properties: it is a hereditary abelian finitary length category. We will define the Ringel-Hall algebra of $\mathscr{H}^{(\rho)}$ and relate it to generalized Kac-Moody Lie algebras. Finally we obtain the Kac type theorem to describe the indecomposable objects in this category, i.e. the indecomposable semi-stable sheaves.