Fedosov quantization in positive characteristic
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- by R. Bezrukavnikov and D. Kaledin;
- J. Amer. Math. Soc. 21 (2008), 409-438
- DOI: https://doi.org/10.1090/S0894-0347-07-00585-1
- Published electronically: November 26, 2007
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Abstract:
We study the problem of deformation quantization for (algebraic) symplectic manifolds over a base field of positive characteristic. We prove a reasonably complete classification theorem for one class of such quantizations; in the course of doing it, we also introduce a notion of a restricted Poisson algebra – the Poisson analog of the standard notion of a restricted Lie algebra – and we prove a version of the Darboux Theorem valid in the positive characteristic setting.References
- R. Bezrukavnikov and D. Kaledin, Fedosov quantization in algebraic context, Mosc. Math. J. 4 (2004), no. 3, 559–592, 782 (English, with English and Russian summaries). MR 2119140, DOI 10.17323/1609-4514-2004-4-3-559-592
- R. V. Bezrukavnikov and D. B. Kaledin, McKay equivalence for symplectic resolutions of quotient singularities, Tr. Mat. Inst. Steklova 246 (2004), no. Algebr. Geom. Metody, Svyazi i Prilozh., 20–42 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 3(246) (2004), 13–33. MR 2101282 [BMR]BMR R. Bezrukavnikov, I. Mirković, and D. Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic, math.RT/0205144.
- Michel Demazure, Lectures on $p$-divisible groups, Lecture Notes in Mathematics, Vol. 302, Springer-Verlag, Berlin-New York, 1972. MR 344261
- Michel Demazure and Pierre Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). Avec un appendice Corps de classes local par Michiel Hazewinkel. MR 302656
- Jean Giraud, Cohomologie non abélienne, Die Grundlehren der mathematischen Wissenschaften, Band 179, Springer-Verlag, Berlin-New York, 1971 (French). MR 344253 [EGA]EGA A. Grothendieck, Éléments de Géométrie Algébrique, III, Publ. Math. IHES 24.
- Maxim Kontsevich, Deformation quantization of algebraic varieties, Lett. Math. Phys. 56 (2001), no. 3, 271–294. EuroConférence Moshé Flato 2000, Part III (Dijon). MR 1855264, DOI 10.1023/A:1017957408559
- James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, NJ, 1980. MR 559531
- Ryszard Nest and Boris Tsygan, Deformations of symplectic Lie algebroids, deformations of holomorphic symplectic structures, and index theorems, Asian J. Math. 5 (2001), no. 4, 599–635. MR 1913813, DOI 10.4310/AJM.2001.v5.n4.a2
- Amnon Yekutieli, Deformation quantization in algebraic geometry, Adv. Math. 198 (2005), no. 1, 383–432. MR 2183259, DOI 10.1016/j.aim.2005.06.009
Bibliographic Information
- R. Bezrukavnikov
- Affiliation: Department of Mathematics, Massachusets Institute of Technology, Cambridge, Massachusetts 02139
- MR Author ID: 347192
- Email: bezrukav@math.mit.edu
- D. Kaledin
- Affiliation: Steklov Institute, Gubkina 8, Moscow, 119991, Russia
- MR Author ID: 634964
- Received by editor(s): October 7, 2005
- Published electronically: November 26, 2007
- Additional Notes: The first author was partially supported by NSF grant DMS-0071967.
The second author was partially supported by CRDF grant RM1-2694-MO05. - © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc. 21 (2008), 409-438
- MSC (2000): Primary 14M99
- DOI: https://doi.org/10.1090/S0894-0347-07-00585-1
- MathSciNet review: 2373355