Finite-dimensional representations of quantum affine algebras at roots of unity

Jonathan Beck; Victor G. Kac

doi:10.1090/s0894-0347-96-00183-x

Finite-dimensional representations of quantum affine algebras at roots of unity

Jonathan Beck
, Victor G. Kac

Harvard University
Massachusetts Institute of Technology

Research output: Contribution to journal › Article › peer-review

30 Scopus citations

Abstract

We describe explicitly the canonical map χ : Spec U_ε(g̃) → Spec Z_ε, where U_ε(g̃) is a quantum loop algebra at an odd root of unity ε. Here Z_ε is the center of U_ε(g̃) and Spec R stands for the set of all finite- dimensional irreducible representations of an algebra R. We show that Spec Z_ε is a Poisson proalgebraic group which is essentially the group of points of G over the regular adeles concentrated at 0 and ∞. Our main result is that the image under χ of Spec U_ε(g̃) is the subgroup of principal adeles.

Original language	English
Pages (from-to)	391-423
Number of pages	33
Journal	Journal of the American Mathematical Society
Volume	9
Issue number	2
DOIs	https://doi.org/10.1090/s0894-0347-96-00183-x
State	Published - Apr 1996
Externally published	Yes

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title = "Finite-dimensional representations of quantum affine algebras at roots of unity",

abstract = "We describe explicitly the canonical map χ : Spec Uε({\~g}) → Spec Zε, where Uε({\~g}) is a quantum loop algebra at an odd root of unity ε. Here Zε is the center of Uε({\~g}) and Spec R stands for the set of all finite- dimensional irreducible representations of an algebra R. We show that Spec Zε is a Poisson proalgebraic group which is essentially the group of points of G over the regular adeles concentrated at 0 and ∞. Our main result is that the image under χ of Spec Uε({\~g}) is the subgroup of principal adeles.",

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AB - We describe explicitly the canonical map χ : Spec Uε(g̃) → Spec Zε, where Uε(g̃) is a quantum loop algebra at an odd root of unity ε. Here Zε is the center of Uε(g̃) and Spec R stands for the set of all finite- dimensional irreducible representations of an algebra R. We show that Spec Zε is a Poisson proalgebraic group which is essentially the group of points of G over the regular adeles concentrated at 0 and ∞. Our main result is that the image under χ of Spec Uε(g̃) is the subgroup of principal adeles.

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Finite-dimensional representations of quantum affine algebras at roots of unity

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