## Duke Mathematical Journal

### 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 $\operatorname{SL}(2,\mathbb{Z})$ of exact autoequivalences of the derived category $D^{b}(\operatorname{Coh}(X))$ 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 $\mathbb{C}[x_{1}^{\pm 1},\ldots,y_{1}^{\pm 1},\ldots]^{\mathfrak{S}_{\infty}}$, that is, the ring of symmetric Laurent polynomials in two sets of countably many variables under the simultaneous symmetric group action.

#### Article information

Source
Duke Math. J., Volume 161, Number 7 (2012), 1171-1231.

Dates
First available in Project Euclid: 4 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1336142074

Digital Object Identifier
doi:10.1215/00127094-1593263

Mathematical Reviews number (MathSciNet)
MR2922373

Zentralblatt MATH identifier
1286.16029

#### Citation

Burban, Igor; Schiffmann, Olivier. On the Hall algebra of an elliptic curve, I. Duke Math. J. 161 (2012), no. 7, 1171--1231. doi:10.1215/00127094-1593263. https://projecteuclid.org/euclid.dmj/1336142074

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