Selective Categories and Linear Canonical Relations
A construction of Wehrheim and Woodward circumvents the problem that compositions of smooth canonical relations are not always smooth, building a category suitable for functorial quantization. To apply their construction to more examples, we introduce a notion of highly selective category, in which...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2014 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2014
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/146541 |
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| Zitieren: | Selective Categories and Linear Canonical Relations / D. Li-Bland, A. Weinstein // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 23 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862640675945709568 |
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| author | Li-Bland, D. Weinstein, A. |
| author_facet | Li-Bland, D. Weinstein, A. |
| citation_txt | Selective Categories and Linear Canonical Relations / D. Li-Bland, A. Weinstein // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 23 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | A construction of Wehrheim and Woodward circumvents the problem that compositions of smooth canonical relations are not always smooth, building a category suitable for functorial quantization. To apply their construction to more examples, we introduce a notion of highly selective category, in which only certain morphisms and certain pairs of these morphisms are ''good''. We then apply this notion to the category SLREL of linear canonical relations and the result WW(SLREL) of our version of the WW construction, identifying the morphisms in the latter with pairs (L,k) consisting of a linear canonical relation and a nonnegative integer. We put a topology on this category of indexed linear canonical relations for which composition is continuous, unlike the composition in SLREL itself. Subsequent papers will consider this category from the viewpoint of derived geometry and will concern quantum counterparts.
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| first_indexed | 2025-12-01T04:10:48Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-146541 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-01T04:10:48Z |
| publishDate | 2014 |
| publisher | Інститут математики НАН України |
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| spelling | Li-Bland, D. Weinstein, A. 2019-02-09T21:06:45Z 2019-02-09T21:06:45Z 2014 Selective Categories and Linear Canonical Relations / D. Li-Bland, A. Weinstein // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53D50; 18F99; 81S10 DOI:10.3842/SIGMA.2014.100 https://nasplib.isofts.kiev.ua/handle/123456789/146541 A construction of Wehrheim and Woodward circumvents the problem that compositions of smooth canonical relations are not always smooth, building a category suitable for functorial quantization. To apply their construction to more examples, we introduce a notion of highly selective category, in which only certain morphisms and certain pairs of these morphisms are ''good''. We then apply this notion to the category SLREL of linear canonical relations and the result WW(SLREL) of our version of the WW construction, identifying the morphisms in the latter with pairs (L,k) consisting of a linear canonical relation and a nonnegative integer. We put a topology on this category of indexed linear canonical relations for which composition is continuous, unlike the composition in SLREL itself. Subsequent papers will consider this category from the viewpoint of derived geometry and will concern quantum counterparts. This paper is a contribution to the Special Issue on Poisson Geometry in Mathematics and Physics. The full
 collection is available at http://www.emis.de/journals/SIGMA/Poisson2014.html.
 In this paper, we will not be discussing the important subject of deformation quantization, in which the
 connection between classical and quantum mechanics is realized by deformations of algebras of observables.
 Alan Weinstein would like to thank the Institut Math´ematique de Jussieu for many years of providing
 a stimulating environment for research. We thank Denis Auroux, Christian Blohmann,
 Sylvain Cappell, Alberto Cattaneo, Pavol Etingof, Theo Johnson-Freyd, Victor Guillemin,
 Thomas Kragh, Jonathan Lorand, Sikimeti Mau, Pierre Schapira, Shlomo Sternberg, Katrin
 Wehrheim, and Chris Woodward for helpful comments on this work. David Li-Bland was supported
 by an NSF Postdoctoral Fellowship DMS-1204779; Alan Weinstein was partially supported
 by NSF Grant DMS-0707137. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Selective Categories and Linear Canonical Relations Article published earlier |
| spellingShingle | Selective Categories and Linear Canonical Relations Li-Bland, D. Weinstein, A. |
| title | Selective Categories and Linear Canonical Relations |
| title_full | Selective Categories and Linear Canonical Relations |
| title_fullStr | Selective Categories and Linear Canonical Relations |
| title_full_unstemmed | Selective Categories and Linear Canonical Relations |
| title_short | Selective Categories and Linear Canonical Relations |
| title_sort | selective categories and linear canonical relations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146541 |
| work_keys_str_mv | AT liblandd selectivecategoriesandlinearcanonicalrelations AT weinsteina selectivecategoriesandlinearcanonicalrelations |