Noncommutative projective curves and quantum loop algebras

Abstract

We show 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 corresponding Kac-Moody algebra. In particular, this yields a geometric realization of the quantized enveloping algebra of elliptic (or $2$-toroidal) algebras of types $D_4^{(1,1)}$, $E^{(1,1)}_6$, $E^{(1,1)}_7$, and $E_{8}^{(1,1)}$ in terms of coherent sheaves on weighted projective lines of genus one or, equivalently, in terms of equivariant coherent sheaves on elliptic curves.