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(1,1)4, E(1,1)6, E(1,1)7, and E(1,1)8 in terms of coherent sheaves on weighted projective lines of genus one or, equivalently, in terms of equivariant coherent sheaves on elliptic curves. © 2004 Duke Mathematical Journal.