On the Hall algebra of an elliptic curve, I

Abstract

We describe the Hall algebra ${\mathbf{H}}_X$ of an elliptic curve $X$ defined over a finite field and show that the group $SL(2,\mathbb{Z})$ of exact autoequivalences of the derived category $D^b(Coh(X\left)\right)$ acts on the Drinfeld double ${\mathbf{DH}}_X$ of ${\mathbf{H}}_X$ by algebra automorphisms. We study a certain natural subalgebra ${\mathbf{U}}_X$ of ${\mathbf{DH}}_X$ 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 $\mathbb{C}[x_1^{\pm 1},\dots ,y_1^{\pm 1},\dots {]}^{{\mathfrak{S}}_{\infty }}$, that is, the ring of symmetric Laurent polynomials in two sets of countably many variables under the simultaneous symmetric group action.