Nonlocality of equivariant star products on \(T^*(\mathbb{R}\mathbb{P}^n)\)
type: Item
instance of: scholarly article
title: Nonlocality of equivariant star products on \(T^*(\mathbb{R}\mathbb{P}^n)\)
zbMATH Open document ID: 1029.53094
DOI: 10.1023/A:1012515230773
author: Ranee Kathryn Brylinski
published in: Letters in Mathematical Physics
publication date: 2004-02-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0010259
review text: The author considers the \(\text{SL}_{n+1}(\mathbb R)\)-invariant star products \(\star_\lambda\) on the smooth functions on \(T^*( \mathbb R \mathbb P^n)\) that are polynomials in the fibre variables which are constructed using the \(\text{SL}_{n+1} (\mathbb R)\)-equivariant symbol calculus defined in [\textit{P. B. A. Lecomte}, and \textit{V. Yu. Ovsienko}, Lett. Math. Phys. 49, 173-196 (1999; Zbl 0989.17015)]. This symbol calculus assigns to every polynomial function \(\phi\) on \(T^*( \mathbb R \mathbb P^n)\) a polynomial \({\mathcal Q}_{\lambda,t}(\phi)\) in the formal parameter \(t\) with values in the differential operators on the space of smooth \(\lambda\)-densities on \(\mathbb R \mathbb P^n\) and the product \(\star_\lambda\) is defined by pulling back the composition of such differential operators with the mapping \({\mathcal Q}_{\lambda,t}\). The aim of the present paper is to show that the so-defined star products are only well-defined on the functions that are polynomials in the fibre variables and cannot be extended to \(C^\infty( T^*( \mathbb R \mathbb P^n))\) and hence do not define star products in the usual sense [cf. \textit{F. Bayen}, \textit{M. Flato}, \textit{C. Frønsdal}, \textit{A. Lichnerowicz} and \textit{D. Sternheimer}, Ann. Phys. 111, 61-151, 111-151 (1978; Zbl 0377.53024 and Zbl 0377.53025)]. It is shown that the reason for this phenomenon is the occurrence of non-local combinations of the fibrewise Euler vector field \(E\) that are only well-defined on polynomials along the fibres. A similar observation of the fact that there are symbolic calculi on cotangent bundles that only yield associative products on the polynomials along the fibres that are not extensible to star products on all smooth functions has also been made in a very simple example on \(T^*\mathbb R\) in [\textit{M. Bordemann}, \textit{N. Neumaier} and \textit{S. Waldmann}, Commun. Math. Phys. 198, 363-396 (1998; Zbl 0968.53056)]. In order to illustrate this nonlocality…
reviewed by: Nikolai Alexander Neumaier
Mathematics Subject Classification ID: 53D55
Mathematics Subject Classification ID: 81S10
Mathematics Subject Classification ID: 16S32
zbMATH DE Number: 1679299
zbMATH Keywords: polynomials
zbMATH Keywords: fibre variables
zbMATH Keywords: \(SL_{n+1}(\mathbb{R})\)-equivariant symbol calculus
MaRDI profile type: Publication
arXiv ID: math/0010259
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description: scientific article; zbMATH DE number 1679299