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Selective Categories and Linear Canonical Relations
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2014, том 10
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Selective Categories and Linear Canonical Relations
Інший ID:
2010 Mathematics Subject Classification: 53D50; 18F99; 81S10
DOI:10.3842/SIGMA.2014.100
DOI:10.3842/SIGMA.2014.100
Посилання:
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.
Підтримка:
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.
Дата:
2014
Переглядів:
521
Завантажень:
263
Анотація:
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.
