On the Hall algebra of an elliptic curve. I.
type: Item
instance of: scholarly article
title: On the Hall algebra of an elliptic curve. I.
zbMATH Open document ID: 1286.16029
DOI: 10.1215/00127094-1593263
author: Igor Burban
author: Olivier Schiffmann
published in: Duke Mathematical Journal
publication date: 2012-06-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0505148
full work available at URL: https://projecteuclid.org/euclid.dmj/1336142074
review text: In this paper the authors give a description of the Hall algebra of an elliptic curve defined over a finite field and show that the group \(\mathrm{SL}(2,\mathbb Z)\) of exact autoequivalences of the bounded derived category of coherent sheaves on \(X\) acts on the Drinfeld double of the Hall algebra by algebra automorphisms. The authors also give a presentation, by generators and relations, of a subalgebra of the Hall algebra which is shown to coincide with Kapranov's spherical Hall algebra and of its Drinfeld double. The latter Drinfeld double is shown to be isomorphic to a flat two-parameter deformation of the ring of symmetric polynomials in two sets of countably many variables under the simultaneous action of the infinite symmetric group.
reviewed by: Volodymyr Mazorchuk
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16T20
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 6047809
zbMATH Keywords: Hall algebras
zbMATH Keywords: elliptic curves
zbMATH Keywords: finite fields
zbMATH Keywords: derived categories
zbMATH Keywords: flat deformations
zbMATH Keywords: symmetric polynomials
zbMATH Keywords: coherent sheaves
zbMATH Keywords: Drinfeld doubles
MaRDI profile type: Publication
OpenAlex ID: W3105029330
arXiv ID: math/0505148
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description: scientific article; zbMATH DE number 6047809