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Conformally Equivariant Quantization - a Complete Classification
Conformally equivariant quantization is a peculiar map between symbols of real weight δ and differential operators acting on tensor densities, whose real weights are designed by λ and λ+δ. The existence and uniqueness of such a map has been proved by Duval, Lecomte and Ovsienko for a generic weight...
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Інститут математики НАН України
2012
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/148414 |
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irk-123456789-1484142019-02-19T01:30:54Z Conformally Equivariant Quantization - a Complete Classification Michel, Jean-Philippe Conformally equivariant quantization is a peculiar map between symbols of real weight δ and differential operators acting on tensor densities, whose real weights are designed by λ and λ+δ. The existence and uniqueness of such a map has been proved by Duval, Lecomte and Ovsienko for a generic weight δ. Later, Silhan has determined the critical values of δ for which unique existence is lost, and conjectured that for those values of δ existence is lost for a generic weight λ. We fully determine the cases of existence and uniqueness of the conformally equivariant quantization in terms of the values of δ and λ. 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 δ, and (ii) in that case the conformally equivariant quantization exists only for a finite number of λ, corresponding to nontrivial conformally invariant differential operators on λ-densities. The assertion (i) is proved in the more general context of IFFT (or AHS) equivariant quantization. 2012 Article Conformally Equivariant Quantization - a Complete Classification / Jean-Philippe Michel // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 25 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53A55; 53A30; 17B56; 47E05 DOI: http://dx.doi.org/10.3842/SIGMA.2012.022 http://dspace.nbuv.gov.ua/handle/123456789/148414 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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Conformally equivariant quantization is a peculiar map between symbols of real weight δ and differential operators acting on tensor densities, whose real weights are designed by λ and λ+δ. The existence and uniqueness of such a map has been proved by Duval, Lecomte and Ovsienko for a generic weight δ. Later, Silhan has determined the critical values of δ for which unique existence is lost, and conjectured that for those values of δ existence is lost for a generic weight λ. We fully determine the cases of existence and uniqueness of the conformally equivariant quantization in terms of the values of δ and λ. 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 δ, and (ii) in that case the conformally equivariant quantization exists only for a finite number of λ, corresponding to nontrivial conformally invariant differential operators on λ-densities. The assertion (i) is proved in the more general context of IFFT (or AHS) equivariant quantization. |
| format |
Article |
| author |
Michel, Jean-Philippe |
| spellingShingle |
Michel, Jean-Philippe Conformally Equivariant Quantization - a Complete Classification Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Michel, Jean-Philippe |
| author_sort |
Michel, Jean-Philippe |
| title |
Conformally Equivariant Quantization - a Complete Classification |
| title_short |
Conformally Equivariant Quantization - a Complete Classification |
| title_full |
Conformally Equivariant Quantization - a Complete Classification |
| title_fullStr |
Conformally Equivariant Quantization - a Complete Classification |
| title_full_unstemmed |
Conformally Equivariant Quantization - a Complete Classification |
| title_sort |
conformally equivariant quantization - a complete classification |
| publisher |
Інститут математики НАН України |
| publishDate |
2012 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/148414 |
| citation_txt |
Conformally Equivariant Quantization - a Complete Classification / Jean-Philippe Michel // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 25 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT micheljeanphilippe conformallyequivariantquantizationacompleteclassification |
| first_indexed |
2023-05-20T17:30:29Z |
| last_indexed |
2023-05-20T17:30:29Z |
| _version_ |
1796153455194144768 |