2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64153We generalize a theorem of Kapranov by showing that the Hall algebra of the category of coherent sheaves on a weighted projective line (over a finite field) provides a realization of the (quantized) enveloping algebra of a certain nilpotent subalgebra of the affinization of the correponding Kac-Moody algebra. In particular this yieds a geometric realization of the quantized enveloping algebra of 2-toroidal (or elliptic) algebras of types D_4, E_6, E_7 or E_8 in terms of weighted projective lines of genus one.Latex, 40 pages, 2 figures, analog of Kac's conjecture added; final version to appearQuantum AlgebraRings and Algebrastext