An extension of the Borel-Weil construction to the quantum group \(U_ q(n)\)
type: Item
instance of: scholarly article
title: An extension of the Borel-Weil construction to the quantum group \(U_ q(n)\)
zbMATH Open document ID: 0757.17009
DOI: 10.1007/BF02097014
author: L. C. Biedenharn
author: M. A. Lohe
published in: Communications in Mathematical Physics
publication date: 1993-01-16T00:00:00Z
review text: The well-known Borel-Weil theorem allows us to construct unitary irreducible representations of a compact Lie group realizing them on holomorphic sections of homogeneous holomorphic line bundles over Kähler manifolds. The aim of this paper is to give the extension of this construction for the quantum group \(U_ q(n)\). The authors employ a recursive approach which differs from the Borel-Weil procedure in that the induction is from an irreducible representation of the quantum subgroup \(U_ q(n-1)\times U_ q(1)\) rather than from a character of the maximal torus \(T_ q\). The base manifold is \(U_ q(n)/(U_ q(n- 1)\times U_ q(1))\) which has a complex structure. The recursive procedure assumes a prior construction of irreducible representations of \(U_ q(n-1)\) by the same method. The extension of the Borel-Weil construction to \(U_ q(n)\) differs in still another way from the procedure actually used in the classical case in that the authors use an analytic method rather than the purely geometric method.
reviewed by: A. U. Klimyk
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 90904
zbMATH Keywords: Borel-Weil theorem
zbMATH Keywords: irreducible representations
zbMATH Keywords: quantum group
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02097014
OpenAlex ID: W2025058578
description: scientific article; zbMATH DE number 90904