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Asymptotic representations and Drinfeld rational fractions

Part of: Lie algebras and Lie superalgebras Groups and algebras in quantum theory

Published online by Cambridge University Press:  10 July 2012

David Hernandez  and
Michio Jimbo
David Hernandez
Affiliation:
Institut de Mathématiques de Jussieu, Université Paris Diderot (Paris VII), 175 rue du Chevaleret, 75013 Paris, France (email: hernandez@math.jussieu.fr)
Michio Jimbo
Affiliation:
Department of Mathematics, Rikkyo University, Toshima-ku, Tokyo 171-8501, Japan (email: jimbomm@rikkyo.ac.jp)
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Abstract

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We introduce and study a category of representations of the Borel algebra associated with a quantum loop algebra of non-twisted type. We construct fundamental representations for this category as a limit of the Kirillov–Reshetikhin modules over the quantum loop algebra and establish explicit formulas for their characters. We prove that general simple modules in this category are classified by n-tuples of rational functions in one variable which are regular and non-zero at the origin but may have a zero or a pole at infinity.

Keywords

quantum affine algebrascategory 𝒪Kirillov–Reshetikhin modulesBorel algebraasymptotic representation theory

MSC classification

Secondary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations 17B10: Representations, algebraic theory (weights) 81R50: Quantum groups and related algebraic methods

Information

Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 5 , September 2012 , pp. 1593 - 1623
Copyright
Copyright © Foundation Compositio Mathematica 2012

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