The quantum loop algebra Uv(Lg) was defined as a generalization of the Drinfeld's new realization of the quantum affine algebra to the loop algebra of any Kac-Moody algebra g. It has been shown by Schiffmann that the Hall algebra of the category of coherent sheaves on a weighted projective line is closely related to the quantum loop algebra Uv(Lg), for some g with a star-shaped Dynkin diagram. In this paper we study Drinfeld's presentation of Uv(Lg) in the double Hall algebra setting, based on Schiffmann's work. We explicitly find out a collection of generators of the double composition algebra DC(Coh(X)) and verify that they satisfy all the Drinfeld relations.
The aim of this paper is to give an alternative proof of Kac's theorem for weighted projective lines over the complex field. The geometric realization of complex Lie algebras arising from derived categories is essentially used.
In this paper, we study the category H (ρ) of semi-stable coherent sheaves of a fixed slope ρ 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 H (ρ) 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.
a α (q) 是维数向量为 α 的绝对不可分解表示同构类的个数, 他还证明了 a α (q) 是 q 的一个整系数多项式, 并猜想 a α (q) ∈ N[q] 且 a α (0) = dim C (g Q) α. Crawley-Boevey 和 Van Den Bergh [4] 证明了, 如果 α 是不可除的 (indivisible),
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