A holomorphic representation of the semidirect sum of symplectic and Heisenberg Lie algebras
type: Item
instance of: scholarly article
title: A holomorphic representation of the semidirect sum of symplectic and Heisenberg Lie algebras
zbMATH Open document ID: 1129.22005
author: Stefan Berceanu
published in: Journal of Geometry and Symmetry in Physics
publication date: 2007-01-02T00:00:00Z
review text: The symplectic group \({\text{Sp}}(2n,{\mathbb{R}})\) may be realized as acting on the Siegel ball, a bounded domain \({\mathcal{D}}_n\) in the space of \(2n\times 2n\) complex matrices whilst the Heisenberg group can be realized as the standard definite hyperquadric in \({\mathbb{C}}^{n+1}\). In this article, the author combines these two realizations to write the semidirect product of these groups as acting on the product \({\mathbb{C}}^n\times{\mathcal{D}}_n\). This action preserves a Kähler metric and can be used to construct unitary representations of the group.
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 32M10
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 81R30
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 5081132
zbMATH Keywords: representation
zbMATH Keywords: symplectic group
zbMATH Keywords: Heisenberg group
reviewed by: M. Eastwood
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5081132