On the Hall algebra of an elliptic curve, I

Abstract

We describe the Hall algebra HX of an elliptic curve X defined over a finite field and show that the group SL(2,Z) of exact autoequivalences of the derived category Db(Coh(X)) acts on the Drinfeld double DHX of HX by algebra automorphisms. We study a certain natural subalgebra UX of DHX for which we give a presentation by generators and relations. This algebra turns out to be a flat two-parameter deformation of the ring of diagonal invariants C[x1±1,…,y1±1,…]S∞, that is, the ring of symmetric Laurent polynomials in two sets of countably many variables under the simultaneous symmetric group action.