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

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

2009-08-12

arxiv.org/abs/0908.1666v1

Mathematics

Representation Theory

29 pages

Scientific paper

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.

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