Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 064, 26 pages      arXiv:0804.4324      https://doi.org/10.3842/SIGMA.2008.064
Contribution to the Special Issue on Deformation Quantization

Hochschild Homology and Cohomology of Klein Surfaces

Frédéric Butin
Université de Lyon, Université Lyon 1, CNRS, UMR5208, Institut Camille Jordan, 43 blvd du 11 novembre 1918, F-69622 Villeurbanne-Cedex, France

Received April 09, 2008, in final form September 04, 2008; Published online September 17, 2008

Abstract
Within the framework of deformation quantization, a first step towards the study of star-products is the calculation of Hochschild cohomology. The aim of this article is precisely to determine the Hochschild homology and cohomology in two cases of algebraic varieties. On the one hand, we consider singular curves of the plane; here we recover, in a different way, a result proved by Fronsdal and make it more precise. On the other hand, we are interested in Klein surfaces. The use of a complex suggested by Kontsevich and the help of Groebner bases allow us to solve the problem.

Key words: Hochschild cohomology; Hochschild homology; Klein surfaces; Groebner bases; quantization; star-products.

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References

  1. Alev J., Farinati M.A., Lambre T., Solotar A. L., Homologie des invariants d'une algèbre de Weyl sous l'action d'un groupe fini, J. Algebra 232 (2000), 564-577.
  2. Alev J., Lambre T., Comparaison de l'homologie de Hochschild et de l'homologie de Poisson pour une déformation des surfaces de Klein, in Algebra and Operator Theory (Tashkent, 1997), Kluwer Acad. Publ., Dordrecht, 1998, 25-38.
  3. Arnold V., Varchenko A., Goussein-Zadé S., Singularités des applications différentiables, première partie, Mir, Moscou, 1986.
  4. Bruguières A., Cattaneo A., Keller B., Torossian C., Déformation, Quantification, Théorie de Lie, Panoramas et Synthèses, SMF, 2005.
  5. Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Physics 111 (1978), 61-110.
    Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., Deformation theory and quantization. II. Physical applications, Ann. Physics 111 (1978), 111-151.
  6. Crawley-Boevey W., Holland M.P., Noncommutative deformations of Kleinian singularities, Duke Math. J. 92 (1998), 605-635.
  7. Chiang L., Chu H., Kang M.C., Generation of invariants, J. Algebra 221 (1999), 232-241.
  8. Fronsdal C., Kontsevich M., Quantization on curves, Lett. Math. Phys. 79 (2007), 109-129, math-ph/0507021.
  9. Gerstenhaber M., The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267-288.
  10. Guieu L., Roger C., avec un appendice de Sergiescu V., L'Algèbre et le Groupe de Virasoro: aspects géométriques et algébriques, généralisations, Publication du Centre de Recherches Mathématiques de Montréal, série "Monographies, notes de cours et Actes de conférences", PM28, 2007.
  11. Kontsevich M., Deformation quantization of Poisson manifolds. I, Preprint IHES, 1997, q-alg/9709040.
  12. Loday J.L., Cyclic homology, Springer-Verlag, Berlin, 1998.
  13. Pichereau A., Cohomologie de Poisson en dimension trois, C. R. Math. Acad. Sci. Paris 340 (2005), 151-154.
  14. Pichereau A., Poisson (co)homology and isolated singularities, J. Algebra 299 (2006), 747-777, math.QA/0511201.
  15. Rannou E., Saux-Picart P., Cours de calcul formel, partie II, éditions Ellipses, 2002.
  16. Roger C., Vanhaecke P., Poisson cohomology of the affine plane, J. Algebra 251 (2002), 448-460.
  17. Springer T.A., Invariant theory, Lecture Notes in Math., Vol. 585, Springer-Verlag, 1977.
  18. Van den Bergh M., Noncommutative homology of some three-dimensional quantum spaces, in Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), K-Theory 8 (1994), 213-230.


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