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SIGMA, 2012, Volume 8, 022, 20 pages (Mi sigma699)  

This article is cited in 6 scientific papers (total in 6 papers)

Conformally equivariant quantization – a complete classification

Jean-Philippe Michel

University of Luxembourg, Campus Kirchberg, Mathematics Research Unit, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg City, Luxembourg

Abstract: Conformally equivariant quantization is a peculiar map between symbols of real weight $\delta$ and differential operators acting on tensor densities, whose real weights are designed by $\lambda$ and $\lambda+\delta$. The existence and uniqueness of such a map has been proved by Duval, Lecomte and Ovsienko for a generic weight $\delta$. Later, Silhan has determined the critical values of $\delta$ for which unique existence is lost, and conjectured that for those values of $\delta$ existence is lost for a generic weight $\lambda$. We fully determine the cases of existence and uniqueness of the conformally equivariant quantization in terms of the values of $\delta$ and $\lambda$. Namely, (i) unique existence is lost if and only if there is a nontrivial conformally invariant differential operator on the space of symbols of weight $\delta$, and (ii) in that case the conformally equivariant quantization exists only for a finite number of $\lambda$, corresponding to nontrivial conformally invariant differential operators on $\lambda$-densities. The assertion (i) is proved in the more general context of IFFT (or AHS) equivariant quantization.

Keywords: quantization, (bi-)differential operators, conformal invariance, Lie algebra cohomology.

DOI: https://doi.org/10.3842/SIGMA.2012.022

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Bibliographic databases:

ArXiv: 1102.4065
MSC: 53A55; 53A30; 17B56; 47E05
Received: July 29, 2011; in final form April 11, 2012; Published online April 15, 2012
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Citation: Jean-Philippe Michel, “Conformally equivariant quantization – a complete classification”, SIGMA, 8 (2012), 022, 20 pp.

Citation in format AMSBIB
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\by Jean-Philippe Michel
\paper Conformally equivariant quantization~-- a~complete classification
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\yr 2012
\vol 8
\papernumber 022
\totalpages 20
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84860855025}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Michel J.-P., “Conformal Geometry of the Supercotangent and Spinor Bundles”, Commun. Math. Phys., 312:2 (2012), 303–336  crossref  mathscinet  zmath  adsnasa  isi  scopus
    2. Silhan J., “Conformally Invariant Quantization - Towards the Complete Classification”, Differ. Geom. Appl., 33:1 (2014), 162–176  crossref  mathscinet  isi  scopus
    3. Michel J.-Ph., “Higher Symmetries of the Laplacian Via Quantization”, Ann. Inst. Fourier, 64:4 (2014), 1581–1609  crossref  mathscinet  zmath  isi  elib  scopus
    4. Michel J.-Ph., “Conformally Equivariant Quantization For Spinning Particles”, Commun. Math. Phys., 333:1 (2015), 261–298  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. Conley Ch.H., Ovsienko V., “Linear Differential Operators on Contact Manifolds”, Int. Math. Res. Notices, 2016, no. 22, 6884–6920  crossref  mathscinet  isi
    6. Ch. H. Conley, D. Grantcharov, “Quantization and injective submodules of differential operator modules”, Adv. Math., 316 (2017), 216–254  crossref  mathscinet  zmath  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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