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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| Дата: | 2013 |
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| Мова: | English |
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Інститут математики НАН України
2013
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/149350 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | 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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irk-123456789-1493502019-02-22T01:24:16Z spo(2|2)-Equivariant Quantizations on the Supercircle S¹|² Mellouli, N. Nibirantiza, A. Radoux, F. 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. 2013 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/149350 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Mellouli, N. Nibirantiza, A. Radoux, F. |
| spellingShingle |
Mellouli, N. Nibirantiza, A. Radoux, F. spo(2|2)-Equivariant Quantizations on the Supercircle S¹|² Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Mellouli, N. Nibirantiza, A. Radoux, F. |
| author_sort |
Mellouli, N. |
| title |
spo(2|2)-Equivariant Quantizations on the Supercircle S¹|² |
| 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 |
| publisher |
Інститут математики НАН України |
| publishDate |
2013 |
| url |
http://dspace.nbuv.gov.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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT melloulin spo22equivariantquantizationsonthesupercircles12 AT nibirantizaa spo22equivariantquantizationsonthesupercircles12 AT radouxf spo22equivariantquantizationsonthesupercircles12 |
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2023-05-20T17:32:47Z |
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2023-05-20T17:32:47Z |
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1796153535746801664 |