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


SIGMA 4 (2008), 060, 17 pages      arXiv:0805.2409      https://doi.org/10.3842/SIGMA.2008.060
Contribution to the Special Issue on Deformation Quantization

Shoikhet's Conjecture and Duflo Isomorphism on (Co)Invariants

Damien Calaque and Carlo A. Rossi
Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland

Received May 23, 2008, in final form August 29, 2008; Published online September 03, 2008

Abstract
In this paper we prove a conjecture of B. Shoikhet. This conjecture states that the tangent isomorphism on homology, between the Poisson homology associated to a Poisson structure on Rd and the Hochschild homology of its quantized star-product algebra, is an isomorphism of modules over the (isomorphic) respective cohomology algebras. As a consequence, we obtain a version of the Duflo isomorphism on coinvariants.

Key words: deformation quantization; formality theorems; cap-products; Duflo isomorphism.

pdf (384 kb)   ps (240 kb)   tex (97 kb)

References

  1. Arnal D., Manchon D., Masmoudi M., Choix des signes pour la formalité de M. Kontsevich, Pacific J. Math. 203, (2002), 23-66, math.QA/0003003.
  2. 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.
  3. Calaque D., Rossi C.A., Lectures on Duflo isomorphisms in Lie algebras and complex geometry, http://math.univ-lyon1.fr/~calaque/LectureNotes/LectETH.pdf.
  4. Calaque D., Rossi C.A., Compatibility with cap-products in Tsygan's formality and homological Duflo isomorphism, arXiv:0805.3444.
  5. Cattaneo A.S., Keller B., Torossian Ch., Bruguières A., Déformation, quantification, théorie de Lie, Panoramas et Synthèses, Vol. 20, SMF, Paris, 2005.
  6. Duflo M., Opérateurs différentiels bi-invariants sur un groupe de Lie, Ann. Sci. École Norm. Sup. (4) 10 (1977), 265-288.
  7. Kontsevich M., Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), 157-216, q-alg/9709040.
  8. Manchon D., Torossian Ch., Cohomologie tangente et cup-produit pour la quantification de Kontsevich, Ann. Math. Blaise Pascal 10 (2003), 75-106, math.QA/0106205.
  9. Pevzner M., Torossian Ch., Isomorphisme de Duflo et la cohomologie tangentielle, J. Geom. Phys. 51 (2004), 487-506, math.QA/0310128.
  10. Shoikhet B., A proof of the Tsygan formality conjecture for chains, Adv. Math. 179 (2003), 7-37, math.QA/0010321.
  11. Shoikhet B., Vanishing of the Kontsevich integrals of the wheels, in EuroConférence Moshé Flato 2000, Part II (Dijon), Lett. Math. Phys. 56, (2001), 141-149, math.QA/0007080.
  12. Tamarkin D., Tsygan B., Cyclic formality and index theorems, EuroConférence Moshé Flato 2000, Part II (Dijon), Lett. Math. Phys. 56 (2001), 85-97.
  13. Tsygan B., Formality conjectures for chains, in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc. Transl. Ser. 2, Vol. 194, Amer. Math. Soc., Providence, RI, 1999, 261-274, math.QA/9904132.


Previous article   Next article   Contents of Volume 4 (2008)