On the Hall algebra of an elliptic curve, I

2005/05/31 by Igor Burban, Olivier Schiffmann · 1 citation
Mathematics · #math.AG #math.QA #msc:17B37 #msc:18E30 #msc:05E05

paper · pdf · doi:10.1215/00127094-1593263

published as Duke Math. J. 161, no. 7 (2012), 1171-1231 · 47 pages, Latex; several changes in the presentation

arxiv created 2009/07/28 · arxiv updated 2019/12/19

Abstract

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[x₁^{± 1}, ..., y₁^{± 1},...]^S_∞, 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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• Cusp eigenforms and the hall algebra of an elliptic curve