Irreducible modules for the quantum affine algebra \(U_{q}(\mathfrak {\widehat{sl}_2})\) and its Borel subalgebra
type: Item
instance of: scholarly article
title: Irreducible modules for the quantum affine algebra \(U_{q}(\mathfrak {\widehat{sl}_2})\) and its Borel subalgebra
zbMATH Open document ID: 1106.17014
author: Paul Terwilliger
author: Georgia Benkart
published in: Journal of Algebra
publication date: 2004-11-29T00:00:00Z
review text: Let \(\mathbb{F}\) be an algebraically closed field of arbitrary characteristic and let \(q\) be an element of \(\mathbb{F}\) that is not a root of unity. The aim of the paper under review is to relate the finite-dimensional irreducible modules of the quantum affine algebra \(U_q(\widehat{\mathfrak{sl}}_2)\) over \(\mathbb{F}\) to the finite-dimensional irreducible modules of its standard Borel subalgebra \(U_q(\mathfrak{b})\). \textit{V. Chari} and \textit{A. Pressley} [Commun. Math. Phys. 142, No. 2, 261--283 (1991; Zbl 0739.17004)] classified the finite-dimensional irreducible modules of \(U_q(\widehat{ \mathfrak{sl}}_2)\) up to isomorphism. Moreover, they showed that every finite-dimensional irreducible modules of \(U_q(\widehat{\mathfrak{sl}}_2)\) is a tensor product of evaluation modules (possibly twisted by an automorphism). The latter are defined via the quantum evaluation homomorphism \(U_q(\widehat{\mathfrak{sl}}_2)\to U_q(\mathfrak{sl}_2)\) introduced by \textit{M. Jimbo} [Lett. Math. Phys. 11, 247--252 (1986; Zbl 0602.17005)]. It should be remarked that the proofs of the paper under review do not use this realization but are quite elementary and require only linear algebra. The main result of the paper establishes a bijection between the isomorphism classes of finite-dimensional irreducible \(U_q(\widehat{\mathfrak{sl}}_2)\)-modules of type \((\varepsilon_0, \varepsilon_1)\) and the isomorphism classes of finite-dimen\-sional irreducible \(\mathcal{B}\)-modules of type \(\alpha\), where \(\mathcal{B}\) is a certain unital associative \(\mathbb{F} \)-algebra, \(\varepsilon_0,\varepsilon_1\in\{\pm 1\}\) such that \(\varepsilon_0\varepsilon_1\) is the eigenvalue of the central element \(K_0K_1\) in \(U_q(\widehat{\mathfrak{sl}}_2)\), and \(\alpha\) is a non-zero element in \(\mathbb{F}\). Most of the paper is devoted to the proof of this result which is an adaptation of a construction of \textit{T. Ito} and the second author in [Tridiagonal pairs and the qu…
reviewed by: Jörg Feldvoss
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 2119560
zbMATH Keywords: quantum affine algebra
zbMATH Keywords: standard Borel subalgebra
zbMATH Keywords: irreducible module
zbMATH Keywords: quantum evaluation homomorphism
zbMATH Keywords: evaluation module
zbMATH Keywords: tensor product
zbMATH Keywords: tridiagonal pair
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.08.016
OpenAlex ID: W1537857615
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DOI: 10.1016/J.JALGEBRA.2004.08.016
description: scientific article; zbMATH DE number 2119560