Second-Order Conformally Equivariant Quantization in Dimension 1|2
This paper is the next step of an ambitious program to develop conformally equivariant quantization on supermanifolds. This problem was considered so far in (super)dimensions 1 and 1|1. We will show that the case of several odd variables is much more difficult. We consider the supercircle S1|2 equip...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2009 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2009
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149129 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Second-Order Conformally Equivariant Quantization in Dimension 1|2 / N. Mellouli // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 14 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862714981785534464 |
|---|---|
| author | Mellouli, N. |
| author_facet | Mellouli, N. |
| citation_txt | Second-Order Conformally Equivariant Quantization in Dimension 1|2 / N. Mellouli // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 14 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | This paper is the next step of an ambitious program to develop conformally equivariant quantization on supermanifolds. This problem was considered so far in (super)dimensions 1 and 1|1. We will show that the case of several odd variables is much more difficult. We consider the supercircle S1|2 equipped with the standard contact structure. The conformal Lie superalgebra K(2) of contact vector fields on S1|2 contains the Lie superalgebra osp(2|2). We study the spaces of linear differential operators on the spaces of weighted densities as modules over osp(2|2). We prove that, in the non-resonant case, the spaces of second order differential operators are isomorphic to the corresponding spaces of symbols as osp(2|2)-modules. We also prove that the conformal equivariant quantization map is unique and calculate its explicit formula.
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| first_indexed | 2025-12-07T17:55:09Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-149129 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:55:09Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Mellouli, N. 2019-02-19T17:34:36Z 2019-02-19T17:34:36Z 2009 Second-Order Conformally Equivariant Quantization in Dimension 1|2 / N. Mellouli // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 14 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 17B10; 17B68; 53D55 https://nasplib.isofts.kiev.ua/handle/123456789/149129 This paper is the next step of an ambitious program to develop conformally equivariant quantization on supermanifolds. This problem was considered so far in (super)dimensions 1 and 1|1. We will show that the case of several odd variables is much more difficult. We consider the supercircle S1|2 equipped with the standard contact structure. The conformal Lie superalgebra K(2) of contact vector fields on S1|2 contains the Lie superalgebra osp(2|2). We study the spaces of linear differential operators on the spaces of weighted densities as modules over osp(2|2). We prove that, in the non-resonant case, the spaces of second order differential operators are isomorphic to the corresponding spaces of symbols as osp(2|2)-modules. We also prove that the conformal equivariant quantization map is unique and calculate its explicit formula. I am grateful to H. Gargoubi and V. Ovsienko for the statement of the problem and constant help. I am also pleased to thank D. Leites for critical reading of this paper and a number of helpful suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Second-Order Conformally Equivariant Quantization in Dimension 1|2 Article published earlier |
| spellingShingle | Second-Order Conformally Equivariant Quantization in Dimension 1|2 Mellouli, N. |
| title | Second-Order Conformally Equivariant Quantization in Dimension 1|2 |
| title_full | Second-Order Conformally Equivariant Quantization in Dimension 1|2 |
| title_fullStr | Second-Order Conformally Equivariant Quantization in Dimension 1|2 |
| title_full_unstemmed | Second-Order Conformally Equivariant Quantization in Dimension 1|2 |
| title_short | Second-Order Conformally Equivariant Quantization in Dimension 1|2 |
| title_sort | second-order conformally equivariant quantization in dimension 1|2 |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149129 |
| work_keys_str_mv | AT melloulin secondorderconformallyequivariantquantizationindimension12 |