RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 SIGMA: Year: Volume: Issue: Page: Find

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

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

Full text: PDF file (471 kB)
Full text: http://emis.mi.ras.ru/.../022
References: PDF file   HTML file

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
Language:

Citation: Jean-Philippe Michel, “Conformally equivariant quantization – a complete classification”, SIGMA, 8 (2012), 022, 20 pp.

Citation in format AMSBIB
\Bibitem{Mic12} \by Jean-Philippe Michel \paper Conformally equivariant quantization~-- a~complete classification \jour SIGMA \yr 2012 \vol 8 \papernumber 022 \totalpages 20 \mathnet{http://mi.mathnet.ru/sigma699} \crossref{https://doi.org/10.3842/SIGMA.2012.022} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2942817} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000303833900001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84860855025} 

• http://mi.mathnet.ru/eng/sigma699
• http://mi.mathnet.ru/eng/sigma/v8/p22

 SHARE:

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
2. Silhan J., “Conformally Invariant Quantization - Towards the Complete Classification”, Differ. Geom. Appl., 33:1 (2014), 162–176
3. Michel J.-Ph., “Higher Symmetries of the Laplacian Via Quantization”, Ann. Inst. Fourier, 64:4 (2014), 1581–1609
4. Michel J.-Ph., “Conformally Equivariant Quantization For Spinning Particles”, Commun. Math. Phys., 333:1 (2015), 261–298
5. Conley Ch.H., Ovsienko V., “Linear Differential Operators on Contact Manifolds”, Int. Math. Res. Notices, 2016, no. 22, 6884–6920
6. Ch. H. Conley, D. Grantcharov, “Quantization and injective submodules of differential operator modules”, Adv. Math., 316 (2017), 216–254
•  Number of views: This page: 183 Full text: 24 References: 23