Minimal affinization of representations of quantum groups: The irregular case
type: Item
instance of: scholarly article
title: Minimal affinization of representations of quantum groups: The irregular case
zbMATH Open document ID: 0857.17011
DOI: 10.1007/BF00943278
author: Vyjayanthi Chari
author: Andrew Pressley
published in: Letters in Mathematical Physics
publication date: 1996-06-04T00:00:00Z
review text: If \({\mathfrak g}\) denotes a finite-dimensional complex simple Lie algebra, and \(V\) a finite-dimensional irreducible representation of \(U_q({\mathfrak g})\), then by definition an affinization \(\widehat{V}\) is an irreducible representation of the quantum affine algebra \(U_q(\widehat{\mathfrak g})\) containing \(V\) with multiplicity one and the property that the highest weights of all other irreducible \(U_q({\mathfrak g})\)-components of \(\widehat{V}\) are strictly smaller than the highest weight \(\lambda\) of \(V\). In this paper the authors deal with the special case that \(U_q({\mathfrak g})\) is of type \(D_4\) and \(\lambda\) is orthogonal to the simple root corresponding to the triple node in the Dynkin diagram of \({\mathfrak g}\). In this ``irregular'' case it is proved, that the number of affinizations, which are minimal with respect to a natural ordering in the set of affinizations, increases with \(\lambda\) and has no upper bound independent of \(\lambda\). This in contrast to the ``regular'' cases in which generally there is a unique minimal affinization up to \(U_q({\mathfrak g})\) isomorphisms. The proof relies on certain non minimal affinizations in the \(A_n\) case, and by the way a conjecture of the authors stated in an earlier paper [ibid. 30, 131-145 (1994; Zbl 0795.17008)] is disproved by a counterexample for \({\mathfrak g}\) of type \(A_2\).
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 874334
zbMATH Keywords: quantum affine algebras
zbMATH Keywords: quantum groups
zbMATH Keywords: representation
zbMATH Keywords: affinizations
reviewed by: Helmut Boseck
MaRDI profile type: Publication
cites work: Braid group action and quantum affine algebras
cites work: Minimal affinizations of representations of quantum groups: The rank 2 case
cites work: Quantum affine algebras
cites work: Small representations of quantum affine algebras
cites work: Q4846800
cites work: Minimal affinizations of representations of quantum groups: The nonsimply-laced case
Recommended article: Minimal affinizations of representations of quantum groups: The nonsimply-laced case
Recommended article: Non-minimality of certain irregular coherent preminimal affinizations
Recommended article: Restricted limits of minimal affinizations
Recommended article: Q4865647
Recommended article: On tensor products of a minimal affinization with an extreme Kirillov-Reshetikhin module for type \(A\)
Recommended article: Q3152491
Recommended article: On minimal affinizations of representations of quantum groups
Recommended article: Q5307721
Recommended article: Beyond Kirillov-Reshetikhin modules
description: scientific article; zbMATH DE number 874334