Noncommutative Projective Curves and Quantum Loop Algebras

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 coherent sheaves on weighted projective lines of genus one. Dedicated to Igor Frenkel on the occasion of his 50th birthday . Introduction. The geometric approach to quantum groups developed in the past 15 years is based on a deep relationship between simple or affine (and to a lesser extent Kac-Moody) Lie algebras and certain finite-dimensional hereditary algebras. More precisely, let g be a Kac-Moody algebra and Γ its Dynkin diagram. Ringel proved in [Ri1] that the Hall algebra of the category of representations, over a finite field Fq, of a quiver whose underlying graph is Γ provides a realization of the “positive part” Uq (g) of the Drinfeld-Jimbo quantum group associated to g. This result was the starting point of Lusztig’s geometric construction of the canonical basis of Uq (g) (see [Lu]). Another natural example of a hereditary category is provided by the category Coh(X) of coherent sheaves on a smooth projective curve X . In the remarkable paper [Kap], Kapranov observed a striking analogy between a function field analog of the algebra of unramified automorphic forms (forGL(N) for allN) and Drinfeld’s loop-like realization of quantum affine algebras. In particular, his result provides an isomorphism between a natural subalgebra of the Hall algebra of the category Coh(P(Fq)) and Drinfeld’s “positive part” Uq (ŝl2). In this paper we extend Kapranov’s result. Rather than considering higher genus smooth projective curves (whose Hall algebras are wild) we study the Hall algebra of the category of coherent sheaves on certain “noncommutative smooth projective curves”the so-called weighted projective lines, introduced by Geigle and Lenzing ([GL]). These fundamental examples of “noncommutative smooth projective curves” have attracted some attention lately (see e.g [Hap2]). Our main result (Theorem 5.2) provides, when g is a simple Lie algebra or an affine Lie algebra of type D (1) 4 , E (1) 6 , E (1) 7 or E (1) 8 , a natural embedding Uq(n̂) →֒ HCoh(Xg) of a certain “positive part” of Uq(ĝ) into the Hall algebra of a suitable weighted projective line Xg. In particular, this gives a geometric construction of the quantum toroidal algebras of type D4, E6, E7 and E8. In this case, the two loops have a natural interpretation as the two discrete invariants (rank and degree) of a coherent sheaf on Xg. Note that the exceptional role played by these four types among all affine Dynkin diagrams is well-known in the theory of quadratic forms (see e.g [Ri3]). Our proof