On the Hall algebra of an elliptic curve, I

Details On the Hall algebra of an elliptic curve, I On the Hall algebra of an elliptic curve, I

2005-05-09

arxiv.org/abs/math/0505148v2

Mathematics

Algebraic Geometry

47 pages, Latex; several changes in the presentation

Scientific paper

In this article we describe the Hall algebra H_X of an elliptic curve X defined over a finite field and show that the group SL(2,Z) of exact auto-equivalences of the derived category D^b(Coh(X)) acts on the Drinfeld double DH_X of H_X by algebra automorphisms. Next, we study a certain natural subalgebra U_X of DH_X 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[x_1^{\pm 1}, ..., y_1^{\pm 1},...]^{S_{\infty}}, i.e. the ring of symmetric Laurent polynomials in two sets of countably many variables under the simultaneous symmetric group action.

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