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

Abstract

In this paper, we study the category ${\mathcal{H}}^{(\rho )}$ 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 ${\mathcal{H}}^{(\rho )}$ 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.

29 pages

References in corpus (1)

• Kac's Theorem for weighted projective lines