## Duke Mathematical Journal

### On the Hall algebra of an elliptic curve, II

Olivier Schiffmann

#### Abstract

The Hall algebra $U^{+}_{E}$ of the category of coherent sheaves on an elliptic curve $E$ defined over a finite field has been explicitly described and shown to be a 2-parameter deformation of the ring of diagonal invariants $R^{+}=C[x_{1}^{\pm1},\ldots,y_{1},\ldots]^{S_{\infty}}$ (in infinitely many variables). We study a geometric version of this Hall algebra, by considering a convolution algebra of perverse sheaves on the moduli spaces of coherent sheaves on $E$. This allows us to define a canonical basis $B$ of $U^{+}_{E}$ in terms of intersection cohomology complexes. We also give a characterization of this basis in terms of an involution, a lattice, and a certain PBW-type basis.

#### Article information

Source
Duke Math. J., Volume 161, Number 9 (2012), 1711-1750.

Dates
First available in Project Euclid: 6 June 2012

https://projecteuclid.org/euclid.dmj/1338987167

Digital Object Identifier
doi:10.1215/00127094-1593362

Mathematical Reviews number (MathSciNet)
MR2922373

Zentralblatt MATH identifier
1253.14018

#### Citation

Schiffmann, Olivier. On the Hall algebra of an elliptic curve, II. Duke Math. J. 161 (2012), no. 9, 1711--1750. doi:10.1215/00127094-1593362. https://projecteuclid.org/euclid.dmj/1338987167

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