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

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