spo(2|2)-Equivariant Quantizations on the Supercircle S¹|²
We consider the space of differential operators Dλμ acting between λ- and μ-densities defined on S¹|² endowed with its standard contact structure. This contact structure allows one to define a filtration on Dλμ which is finer than the classical one, obtained by writting a differential operator in te...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2013 |
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Інститут математики НАН України
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| Zitieren: | spo(2|2)-Equivariant Quantizations on the Supercircle S¹|² / N. Mellouli, A. Nibirantiza, F. Radoux // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 27 назв. — англ. |
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Mellouli, N. Nibirantiza, A. Radoux, F. 2019-02-21T07:08:09Z 2019-02-21T07:08:09Z 2013 spo(2|2)-Equivariant Quantizations on the Supercircle S¹|² / N. Mellouli, A. Nibirantiza, F. Radoux // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53D10; 17B66; 17B10 DOI: http://dx.doi.org/10.3842/SIGMA.2013.055 https://nasplib.isofts.kiev.ua/handle/123456789/149350 We consider the space of differential operators Dλμ acting between λ- and μ-densities defined on S¹|² endowed with its standard contact structure. This contact structure allows one to define a filtration on Dλμ which is finer than the classical one, obtained by writting a differential operator in terms of the partial derivatives with respect to the different coordinates. The space Dλμ and the associated graded space of symbols Sδ (δ=μ−λ) can be considered as spo(2|2)-modules, where spo(2|2) is the Lie superalgebra of contact projective vector fields on S¹|². We show in this paper that there is a unique isomorphism of spo(2|2)-modules between Sδ and Dλμ that preserves the principal symbol (i.e. an spo(2|2)-equivariant quantization) for some values of δ called non-critical values. Moreover, we give an explicit formula for this isomorphism, extending in this way the results of [Mellouli N., SIGMA 5 (2009), 111, 11 pages] which were established for second-order differential operators. The method used here to build the spo(2|2)-equivariant quantization is the same as the one used in [Mathonet P., Radoux F., Lett. Math. Phys. 98 (2011), 311-331] to prove the existence of a pgl(p+1|q)-equivariant quantization on Rp|q. It is a pleasure to thank T. Leuther, P. Mathonet, J.-P. Michel and V. Ovsienko for numerous fruitful discussions and for their interest in our work. We also warmly thank the referees for their suggestions and remarks which considerably improved the paper. This research has been funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications spo(2|2)-Equivariant Quantizations on the Supercircle S¹|² Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
spo(2|2)-Equivariant Quantizations on the Supercircle S¹|² |
| spellingShingle |
spo(2|2)-Equivariant Quantizations on the Supercircle S¹|² Mellouli, N. Nibirantiza, A. Radoux, F. |
| title_short |
spo(2|2)-Equivariant Quantizations on the Supercircle S¹|² |
| title_full |
spo(2|2)-Equivariant Quantizations on the Supercircle S¹|² |
| title_fullStr |
spo(2|2)-Equivariant Quantizations on the Supercircle S¹|² |
| title_full_unstemmed |
spo(2|2)-Equivariant Quantizations on the Supercircle S¹|² |
| title_sort |
spo(2|2)-equivariant quantizations on the supercircle s¹|² |
| author |
Mellouli, N. Nibirantiza, A. Radoux, F. |
| author_facet |
Mellouli, N. Nibirantiza, A. Radoux, F. |
| publishDate |
2013 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We consider the space of differential operators Dλμ acting between λ- and μ-densities defined on S¹|² endowed with its standard contact structure. This contact structure allows one to define a filtration on Dλμ which is finer than the classical one, obtained by writting a differential operator in terms of the partial derivatives with respect to the different coordinates. The space Dλμ and the associated graded space of symbols Sδ (δ=μ−λ) can be considered as spo(2|2)-modules, where spo(2|2) is the Lie superalgebra of contact projective vector fields on S¹|². We show in this paper that there is a unique isomorphism of spo(2|2)-modules between Sδ and Dλμ that preserves the principal symbol (i.e. an spo(2|2)-equivariant quantization) for some values of δ called non-critical values. Moreover, we give an explicit formula for this isomorphism, extending in this way the results of [Mellouli N., SIGMA 5 (2009), 111, 11 pages] which were established for second-order differential operators. The method used here to build the spo(2|2)-equivariant quantization is the same as the one used in [Mathonet P., Radoux F., Lett. Math. Phys. 98 (2011), 311-331] to prove the existence of a pgl(p+1|q)-equivariant quantization on Rp|q.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/149350 |
| citation_txt |
spo(2|2)-Equivariant Quantizations on the Supercircle S¹|² / N. Mellouli, A. Nibirantiza, F. Radoux // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 27 назв. — англ. |
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AT melloulin spo22equivariantquantizationsonthesupercircles12 AT nibirantizaa spo22equivariantquantizationsonthesupercircles12 AT radouxf spo22equivariantquantizationsonthesupercircles12 |
| first_indexed |
2025-12-07T15:38:31Z |
| last_indexed |
2025-12-07T15:38:31Z |
| _version_ |
1850864473025282048 |