Covariant deformations on polarized orbits of a Lie group
type: Item
instance of: scholarly article
title: Covariant deformations on polarized orbits of a Lie group
zbMATH Open document ID: 0834.22015
DOI: 10.1016/0393-0440(94)90039-6
published in: Journal of Geometry and Physics
publication date: 1996-03-05T00:00:00Z
review text: Let \({\mathcal O}\) be a coadjoint orbit in the dual of the Lie algebra of a real Lie group, with its natural structure of a symplectic manifold. The aim is to construct a star-product on \({\mathcal O}\): \[ u*v = C_0 (u,v) + tC_1 (u,v) + t^2 C_2 (u,v) + \cdots \] where \(u,v\) are smooth functions on \({\mathcal O},t\) is the deformation parameter, \(C_0(u,v) = uv\) is the ordinary product and \(C_1 (u,v) = \{u,v\}\) is the Poisson bracket. Star-products are known to exist on any symplectic manifold, but here it is natural to ask for a covariant star-product, i.e. \(u*v - v*u = 2t \{u,v\}\) whenever \(u(\xi) = \langle \xi,X \rangle\) and \(v(\xi) = \langle \xi, Y \rangle \) are coordinate functions associated to elements \(X,Y\) in the Lie algebra. -- The main result is the existence of such a covariant star-product on the orbit \({\mathcal O}\), as soon as it admits a (real or complex) polarization. Its construction is obtained by means of suitably chosen coordinates on a Kähler manifold naturally associated to this geometric setting.
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 58H15
Mathematics Subject Classification ID: 14D15
zbMATH DE Number: 700988
zbMATH Keywords: coadjoint orbit
zbMATH Keywords: Lie algebra
zbMATH Keywords: real Lie group
zbMATH Keywords: star-product
zbMATH Keywords: deformation parameter
zbMATH Keywords: polarization
zbMATH Keywords: Kähler manifold
author: D. Arnal
author: Jean Ludwig
author: M. Masmoudi
reviewed by: François Rouvière
MaRDI profile type: Publication
full work available at URL: 0393-0440(94)90039-6
OpenAlex ID: W2035907796
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description: scientific article; zbMATH DE number 700988