Duke Mathematical Journal

On the Hall algebra of an elliptic curve, I

Igor Burban and Olivier Schiffmann

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Abstract

We describe the Hall algebra HX of an elliptic curve X defined over a finite field and show that the group SL(2,Z) of exact autoequivalences of the derived category Db(Coh(X)) acts on the Drinfeld double DHX of HX by algebra automorphisms. We study a certain natural subalgebra UX of DHX 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, 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

Subjects
Primary: 16T05: Hopf algebras and their applications [See also 16S40, 57T05]
Secondary: 22E65: Infinite-dimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05]

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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