Quantization of canonical cones of algebraic curves

Benjamin Enriquez[1]; Alexander Odesskii[2]

  • [1] Université Louis Pasteur, IRMA, 7 rue René Descartes, 67084 Strasbourg Cedex (France)
  • [2] Landau Institute of Theoretical Physics, 2 Kosygina str., 117334 Moscow (Russie)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 6, page 1629-1663
  • ISSN: 0373-0956

Abstract

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We introduce a quantization of the graded algebra of functions on the canonical cone of an algebraic curve C , based on the theory of formal pseudodifferential operators. When C is a complex curve with Poincaré uniformization, we propose another, equivalent construction, based on the work of Cohen-Manin-Zagier on Rankin-Cohen brackets. We give a presentation of the quantum algebra when C is a rational curve, and discuss the problem of constructing algebraically “differential liftings”.

How to cite

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Enriquez, Benjamin, and Odesskii, Alexander. "Quantization of canonical cones of algebraic curves." Annales de l’institut Fourier 52.6 (2002): 1629-1663. <http://eudml.org/doc/116022>.

@article{Enriquez2002,
abstract = {We introduce a quantization of the graded algebra of functions on the canonical cone of an algebraic curve $C$, based on the theory of formal pseudodifferential operators. When $C$ is a complex curve with Poincaré uniformization, we propose another, equivalent construction, based on the work of Cohen-Manin-Zagier on Rankin-Cohen brackets. We give a presentation of the quantum algebra when $C$ is a rational curve, and discuss the problem of constructing algebraically “differential liftings”.},
affiliation = {Université Louis Pasteur, IRMA, 7 rue René Descartes, 67084 Strasbourg Cedex (France); Landau Institute of Theoretical Physics, 2 Kosygina str., 117334 Moscow (Russie)},
author = {Enriquez, Benjamin, Odesskii, Alexander},
journal = {Annales de l’institut Fourier},
keywords = {algebraic curves; canonical cones; formal pseudodifferential operators; Rankin-Cohen brackets; Poincaré uniformization},
language = {eng},
number = {6},
pages = {1629-1663},
publisher = {Association des Annales de l'Institut Fourier},
title = {Quantization of canonical cones of algebraic curves},
url = {http://eudml.org/doc/116022},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Enriquez, Benjamin
AU - Odesskii, Alexander
TI - Quantization of canonical cones of algebraic curves
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 6
SP - 1629
EP - 1663
AB - We introduce a quantization of the graded algebra of functions on the canonical cone of an algebraic curve $C$, based on the theory of formal pseudodifferential operators. When $C$ is a complex curve with Poincaré uniformization, we propose another, equivalent construction, based on the work of Cohen-Manin-Zagier on Rankin-Cohen brackets. We give a presentation of the quantum algebra when $C$ is a rational curve, and discuss the problem of constructing algebraically “differential liftings”.
LA - eng
KW - algebraic curves; canonical cones; formal pseudodifferential operators; Rankin-Cohen brackets; Poincaré uniformization
UR - http://eudml.org/doc/116022
ER -

References

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  1. M. Adler, On a trace functional for formal pseudo-differential operators and the symplectic structure of the KdV equation, Invent. Math 50 (1979), 219-248 Zbl0393.35058MR520927
  2. A. Beauville, Systèmes hamiltoniens complètement intégrables associés aux surfaces K3, Problems in the theory of surfaces and their classification (Cortona, 1988) XXXII (1991), 25-31, Academic Press Zbl0827.58022
  3. P. Beazley Cohen, Yu. Manin, D. Zagier, Automorphic pseudodifferential operators, paper in memory of Irene Dorfman, Algebraic aspects of integrable systems 26 (1997), 17-47, Birkhäuser Boston, Boston, MA Zbl1055.11514
  4. L. Boutet de Monvel, Complex star algebras, (1999), 1-27, Kluwer Acad. Publishers, the Netherlands Zbl0980.53106
  5. B. Feigin, A. Odesskii, Sklyanin's elliptic algebras, Functional Anal. Appl 23 (1990), 207-214 Zbl0713.17009MR1026987
  6. P. Griffiths, J. Harris, Principles of algebraic geometry, (1994), J. Wiley and Sons, Inc., New York Zbl0836.14001MR1288523
  7. J. Harris, Algebraic geometry. A first course, 133 (1985), Springer-Verlag, New York Zbl0779.14001MR1182558
  8. M. Kontsevich, Deformation quantization of Poisson manifolds, I Zbl1058.53065
  9. Y. Manin, Algebraic aspects of differential equations, J. Sov. Math 11 (1979), 1-128 Zbl0419.35001
  10. A. Odesskii, V. Rubtsov, Polynomial Poisson algebras with regular structure of symplectic leaves, (2001) Zbl1138.53314
  11. V. Ovsienko, Exotic deformation quantization, J. Differential Geom 45 (1997), 390-406 Zbl0879.58028MR1449978

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