Duke Mathematical Journal

Noncommutative projective curves and quantum loop algebras

Olivier Schiffmann

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Abstract

We show that the Hall algebra of the category of coherent sheaves on a weighted projective line over a finite field provides a realization of the (quantized) enveloping algebra of a certain nilpotent subalgebra of the affinization of the corresponding Kac-Moody algebra. In particular, this yields a geometric realization of the quantized enveloping algebra of elliptic (or $2$-toroidal) algebras of types $D_4^{(1,1)}$, $E^{(1,1)}_6$, $E^{(1,1)}_7$, and $E_{8}^{(1,1)}$ in terms of coherent sheaves on weighted projective lines of genus one or, equivalently, in terms of equivariant coherent sheaves on elliptic curves.

Article information

Source
Duke Math. J., Volume 121, Number 1 (2004), 113-168.

Dates
First available in Project Euclid: 21 December 2003

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1072058751

Digital Object Identifier
doi:10.1215/S0012-7094-04-12114-1

Mathematical Reviews number (MathSciNet)
MR2031167

Zentralblatt MATH identifier
1054.17021

Subjects
Primary: 22E
Secondary: 16G 18F

Citation

Schiffmann, Olivier. Noncommutative projective curves and quantum loop algebras. Duke Math. J. 121 (2004), no. 1, 113--168. doi:10.1215/S0012-7094-04-12114-1. https://projecteuclid.org/euclid.dmj/1072058751


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