Unitary positive energy representations of the gauge group
type: Item
instance of: scholarly article
title: Unitary positive energy representations of the gauge group
zbMATH Open document ID: 0618.22013
DOI: 10.1007/BF00570762
published in: Letters in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: Let G be a group of smooth functions from a Riemannian manifold (the space-time) M into the compact Lie group K, and \({\mathfrak g}\) its Lie algebra. The author constructs the generalized Verma modules of \({\mathfrak g}\). The so-called elementary and semi-elementary representations of \({\mathfrak g}\) are introduced by means of these modules. Let \(\omega\) be the involutive antilinear antiautomorphism of \({\mathfrak g}\). A representation \(\pi\) of \({\mathfrak g}\) is said to be unitarizable with respect to \(\omega\) if there exists a positive definite contravariant Hermitian form H, such that \(H(\pi (g)u,v)=H(u,\pi (\omega g)v)\) for every \(g\in {\mathfrak g}\). It is proved that the only unitarizable representations of \({\mathfrak g}\) with highest weight are either elementary or semi-elementary ones. The algebra \({\mathfrak g}\) does not admit faithful unitarizable representations with highest weight.
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 17B65
zbMATH DE Number: 4002406
zbMATH Keywords: generalized loop algebras
zbMATH Keywords: gauge group
zbMATH Keywords: generalized Verma modules
zbMATH Keywords: unitarizable representations
zbMATH Keywords: representations with highest weight
author: Bruno Torrésani
MaRDI profile type: Publication
cites work: Q3693672
cites work: Irreducibility and reducibility for the energy representation of the group of mappings of a Riemannian manifold into a compact semisimple Lie group
full work available at URL: https://doi.org/10.1007/bf00570762
OpenAlex ID: W2021657933
description: scientific article; zbMATH DE number 4002406