Universal overgroup
type: Item
instance of: scholarly article
title: Universal overgroup
zbMATH Open document ID: 1213.37086
author: D. Arnal
author: Mohamed Selmi
author: Amel Zergane
published in: Journal of Geometry and Physics
publication date: 2011-01-14T00:00:00Z
review text: It is well known that to any unitary representation \(\pi\) of a Lie group \(G\) on a finite-dimensional complex vector space \(V\) one can associate the moment map of the corresponding Hamiltonian action on the complex projective space \(PV\). This concept is extended for infinite-dimensional representations \(\pi\) on a Hilbert space \((H,(\cdot|\cdot))\) in \textit{N. J. Wildberger} [Invent. Math. 98, No.~2, 281--292 (1989; Zbl 0684.22005)]. Here, instead of \(H\) one should consider a subspace \(V\) of smooth vectors for a representation \(\pi\). The range \(l_\pi\) of the moment map does not characterize the representation \(\pi\). Following \textit{D. Arnal} and \textit{M. Selmi} [Bull. Sci. Math. 132, No.~1, 54--69 (2008; Zbl 1138.22006)], a way to separate irreducible unitary representations \(\pi\) for a Lie group \(G\) by moment sets is to use an overgroup \(\widetilde G\) and extensions of each representation \(\pi\) to a representation \(\widetilde {\pi}\) of \(\widetilde G\). In this paper the construction of a universal overgroup \(\widetilde G\) is proposed. It is an infinite-dimensional Fréchet-Lie group and extend each \(\pi\) to a Hamiltonian action \(\widetilde{\pi}\) of \(\tilde G\), such that the moment set of \(\widetilde{\pi}\) characterizes \(\pi\). The authors use the following interesting geometric construction. They define a natural Hamiltonian action \(\lambda\) of a Fréchet-Lie group \(G \times V\) (semidirect product with a real abelian group \(V\)) on \(V\) and prove that the moment set \(\ell\) of \(\lambda\) is the graph of the map \(\theta: V\to \text{Lie}(G)^*\), \(\theta(v)(x)= \text{Im}(\pi'(x)v|v)\). The projection of the moment set \(\ell\) on the dual \(\text{Lie}(G)^*\) is the positive cone supported by the the usual moment set associated to the representation \(\pi\). Here \(\pi'\) is the associated representation of the Lie algebra \(\text{Lie}(G)\).
Mathematics Subject Classification ID: 37J15
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 22D30
zbMATH DE Number: 5836605
zbMATH Keywords: moment map
zbMATH Keywords: representations of Lie groups
zbMATH Keywords: Fréchet manifold
zbMATH Keywords: semi-direct product
zbMATH Keywords: universal overgroup
MaRDI profile type: Publication
full work available at URL: j.geomphys.2010.09.015
OpenAlex ID: W4205471207
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DOI: 10.1016/J.GEOMPHYS.2010.09.015
description: scientific article; zbMATH DE number 5836605