This is a joint work with Rujing Dou and Jie Xiao. We are interested in
the category of coherent sheaves of a fixed slope on a weighted
projective curve defined over a finite field. More generally we
consider a hereditary abelian finitary length category A. We show that
the double composition Hall algebra associated to A is isomorphic to
the quantized enveloping algebra of a generalized Kac-Moody algebra,
and the whole double Hall algebra is isomorphic to the quotient of the
quantized enveloping algebra of a larger generalized Kac-Moody algebra,
factoring out an ideal belonging to the Cartan part. Both Lie algebras
are developed from the Euler form of the category A. As an application,
we obtain the classification of the dimension vectors of the
indecomposable semistable coherent sheaves of the fixed slope, via the
positive roots of the Lie algebra (i.e. an analogue of Kac's theorem).