Hall Algebras On The Categories Of Coherent Sheaves On Weighted Projective Lines

We study the Hall algebras on the categories of weighted projective lines whichhave links with loop algebras and Kac-Moody algebras. The main results we obtainedare as following:First, we study the category H (ρ) of semi-stable coherent sheaves of a fixed slopeρover a weighted projective curve. This category has nice properties: it is a heredi-tary abelian finitary length category. We will define the Ringel-Hall algebra of H (ρ)and relate it to the generalized Kac-Moody algebras. Finally we obtain the Kac typetheorem to describe the indecomposable objects in this category, i.e., give the corre-spondence between the indecomposable objects and the root system of the generalizedKac-Moody algebra.Secondly, the quantum loop algebra Uv(?g) was defined as a generalization ofthe Drinfeld's new realization of quantum a?ne algebra to the loop algebra of anyKac-Moody algebra g. Schi?mann[18] has proved ( and conjectured ) that the Hallalgebra of the category of coherent sheaves over weighted projective lines providesa realization of Uv(?g) for those g associated to a star-shaped Dynkin diagram. Inthis paper we explicitly find out the elements in the Hall algebra H(Coh(X)) satisfyingpart of Drinfeld's relations, as addition to Schi?mann's work. Further we verify allDrinfeld's relations in the double Hall algebra DH(Coh(X)). As a corollary, we deducethat the double composition algebra is isomorphic to the whole quantum loop algebrawhen g is of finite or a?ne type.Thirdly, we give an alternative proof of Kac's theorem for weighted projectivelines ([20]) over the complex field. The geometric realization of complex Lie algebrasarising form derived categories ([21]) is essentially used.