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

Authors: Jonathan Beck and Victor G. Kac
Journal: J. Amer. Math. Soc. 9 (1996), 391-423
MSC (1991): Primary 17B37, 81R50
MathSciNet review: 1317228
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Abstract: We describe explicitly the canonical map $\chi :$ Spec $U_{\varepsilon }(\tilde {\mathfrak {g}}) \rightarrow$ Spec $Z_{\varepsilon }$, where $U_{\varepsilon } (\tilde {\mathfrak {g}})$ is a quantum loop algebra at an odd root of unity $\varepsilon$. Here $Z_{\varepsilon }$ is the center of $U_{\varepsilon }(\tilde {\mathfrak {g}})$ and Spec $R$ stands for the set of all finite–dimensional irreducible representations of an algebra $R$. We show that Spec $Z_{\varepsilon }$ is a Poisson proalgebraic group which is essentially the group of points of $G$ over the regular adeles concentrated at $0$ and $\infty$. Our main result is that the image under $\chi$ of Spec $U_{\varepsilon }(\tilde {\mathfrak {g}})$ is the subgroup of principal adeles.

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Additional Information

Jonathan Beck
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Victor G. Kac
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received by editor(s): October 28, 1994
Received by editor(s) in revised form: November 9, 1994
Additional Notes: The first author was supported by an NSF Postdoctoral Fellowship.
The second author was supported in part by NSF grant DMS–9103792.
Article copyright: © Copyright 1996 American Mathematical Society