Quantization and Morita equivalence for constant Dirac structures on tori

Xiang Tang; Alan Weinstein[1]

  • [1] University of California, Department of Mathematics, Berkeley, CA 94720 (USA), University of California, Department of Mathematics Davis, CA 95616 (USA)

Annales de l’institut Fourier (2004)

  • Volume: 54, Issue: 5, page 1565-1580
  • ISSN: 0373-0956

Abstract

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We define a C * -algebraic quantization of constant Dirac structures on tori and prove that O ( n , n | ℤ ) -equivalent structures have Morita equivalent quantizations. This completes and extends from the Poisson case a theorem of Rieffel and Schwarz.

How to cite

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Tang, Xiang, and Weinstein, Alan. "Quantization and Morita equivalence for constant Dirac structures on tori." Annales de l’institut Fourier 54.5 (2004): 1565-1580. <http://eudml.org/doc/116152>.

@article{Tang2004,
abstract = {We define a $C^*$-algebraic quantization of constant Dirac structures on tori and prove that $O(n,n|\mathbb \{Z\})$-equivalent structures have Morita equivalent quantizations. This completes and extends from the Poisson case a theorem of Rieffel and Schwarz.},
affiliation = {University of California, Department of Mathematics, Berkeley, CA 94720 (USA), University of California, Department of Mathematics Davis, CA 95616 (USA)},
author = {Tang, Xiang, Weinstein, Alan},
journal = {Annales de l’institut Fourier},
keywords = {Dirac structure; Poisson structure; Morita equivalence; quantization},
language = {eng},
number = {5},
pages = {1565-1580},
publisher = {Association des Annales de l'Institut Fourier},
title = {Quantization and Morita equivalence for constant Dirac structures on tori},
url = {http://eudml.org/doc/116152},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Tang, Xiang
AU - Weinstein, Alan
TI - Quantization and Morita equivalence for constant Dirac structures on tori
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 5
SP - 1565
EP - 1580
AB - We define a $C^*$-algebraic quantization of constant Dirac structures on tori and prove that $O(n,n|\mathbb {Z})$-equivalent structures have Morita equivalent quantizations. This completes and extends from the Poisson case a theorem of Rieffel and Schwarz.
LA - eng
KW - Dirac structure; Poisson structure; Morita equivalence; quantization
UR - http://eudml.org/doc/116152
ER -

References

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  1. J. Block, E. Getzler, Quantization of foliations, Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, (New York, 1991) Vol. 1, 2 (1992), 471-487, World Scientific, River Edge, NJ Zbl0812.58028
  2. A. Connes, Noncommutative Geometry, (1994), Academic Press, San Diego Zbl0818.46076MR1303779
  3. A. Connes, M.R. Douglas, A. Schwarz, Noncommutative Geometry and Matrix Theory: Compactification on Tori, J. High Energy Phys (1998) Zbl1018.81052MR1613978
  4. T.J. Courant, Dirac manifolds, Trans. A.M.S 319 (1990), 631-661 Zbl0850.70212MR998124
  5. G. A. Elliott, On the K-theory of the C * algebras generated by a projective representation of a torsion-free discrete abelian group, Operator Algebras and Group Representations (1984), 157-184, Pitman, London Zbl0542.46030
  6. G.A. Elliott, H. Li, Morita equivalence of smooth noncommutative tori Zbl1137.46030
  7. H. Kajiura, Kronecker foliation, D 1 -branes and Morita equivalence of noncommutative two-tori, J. High Energy Phys 8 (2002) Zbl1226.81097MR1942142
  8. M. Kontsevich, Homological algebra of mirror symmetry., Proceedings of the International Congress of Mathematicians (Zürich, 1994) Vol. 1, 2 (1995), 120-139, Birkhäuser, Basel Zbl0846.53021
  9. H. Li, Strong Morita equivalence of higher-dimensional noncommutative tori Zbl1063.46057MR2099203
  10. F. Lizzi, R. Szabo, Noncommutative Geometry and String Duality, Conf. Proc. Corfu Summer Institute on Elementary Particle Physics (Kerkyra, 1998) (1999), Trieste Zbl0994.81111
  11. P.S. Muhly, J.N. Renault, D.P. Williams, Equivalence and isomorphism for groupoid C * -algebras, J. Operator Theory 17 (1987), 3-22 Zbl0645.46040MR873460
  12. S. Mukai, Duality between D ( X ) and D ( X ^ ) with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153-175 Zbl0417.14036MR607081
  13. M.A. Rieffel, Morita equivalence for C * -algebras and W * -algebras, J. Pure Appl. Algebra 5 (1974), 51-96 Zbl0295.46099MR367670
  14. M.A. Rieffel, C * -algebras associated with irrational rotations, Pacific. J. Math. 93 (1981), 415-429 Zbl0499.46039MR623572
  15. M.A. Rieffel, Projective modules over higher-dimensional non-commutative noncommutative tori, Canadian J. Math 40 (1988), 257-338 Zbl0663.46073MR941652
  16. M.A. Rieffel, Deformation quantization of Heisenberg manifolds, Commun. Math. Phys 122 (1989), 531-562 Zbl0679.46055MR1002830
  17. M.A. Rieffel, A. Schwarz, Morita equivalence of multidimensional noncommutative tori, Int. J. Math 10 (1999), 289-299 Zbl0968.46060MR1687145
  18. A. Schwarz, Morita equivalence and duality, Lett. Math. Phys 50 (1999), 309-321 Zbl0967.58004MR1663471
  19. X. Tang, Deformation Quantization of Pseudo Symplectic (Poisson) Groupoids Zbl1119.53061
  20. A. Weinstein, Symplectic groupoids, geometric quantization, and irrational rotation algebras, Symplectic geometry, groupoids, and integrable systems, Séminaire sud-Rhodanien de géométrie à Berkeley (1989) (1991), 281-290, Springer Zbl0731.58031
  21. P. Xu, Noncommutative Poisson algebras, Amer. J. Math 116 (1994), 101-125 Zbl0797.58012MR1262428

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