Duke Mathematical Journal

Hochschild cohomology of the Weyl algebra and traces in deformation quantization

Boris Feigin, Giovanni Felder, and Boris Shoikhet

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We give a formula for a cocycle generating the Hochschild cohomology of the Weyl algebra with coefficients in its dual. It is given by an integral over the configuration space of ordered points on a circle. Using this formula and a noncommutative version of formal geometry, we obtain an explicit expression for the canonical trace in deformation quantization of symplectic manifolds.

Article information

Source
Duke Math. J., Volume 127, Number 3 (2005), 487-517.

Dates
First available in Project Euclid: 18 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1113847337

Digital Object Identifier
doi:10.1215/S0012-7094-04-12733-2

Mathematical Reviews number (MathSciNet)
MR2132867

Zentralblatt MATH identifier
1106.53055

Subjects
Primary: 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
Secondary: 53D55: Deformation quantization, star products

Citation

Feigin, Boris; Felder, Giovanni; Shoikhet, Boris. Hochschild cohomology of the Weyl algebra and traces in deformation quantization. Duke Math. J. 127 (2005), no. 3, 487--517. doi:10.1215/S0012-7094-04-12733-2. https://projecteuclid.org/euclid.dmj/1113847337


Export citation

References

  • \lccF. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Deformation theory and quantization, I, II, Ann. Physics 111 (1977), 61–110; 111–151. ;
  • \lccH. Cartan, “La transgression dans un groupe de Lie et dans un espace fibré principal” in Colloque de topologie: Espaces fibrés (Brussels, 1950), Masson, Paris, 1951, 57–71; reprinted in Oeuvres, Vol. III, Springer, Berlin, 1979, 1268–1283. ;
  • ––––, “Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie” in Colloque de topologie: Espaces fibrés (Brussels, 1950), Masson, Paris, 1951, 15–27; reprinted in Ouevres, Vol. III, Springer, Berlin, 1979, 1255–1267. ;
  • \lccA. Connes, M. Flato, and D. Sternheimer, Closed star products and cyclic cohomology, Lett. Math. Phys. 24 (1992), 1–12.
  • \lccB. Fedosov, Deformation Quantization and Index Theory, Math. Top. 9, Akademie, Berlin, 1996.
  • \lccB. L. Feĭgin and B. L. Tsygan, Cohomology of the Lie algebras of generalized Jacobi matrices (in Russian), Funktsional. Anal. i Prilozhen. 17, no. 2 (1983), 86–87; English translation in Funct. Anal. Appl. 17, no. 2 (1983), 153–155.
  • ––––, “Riemann-Roch theorem and Lie algebra cohomology, I” in Proceedings of the Winter School on Geometry and Physics (Srní, Czech Rep., 1988), Rend. Circ. Mat. Palermo (2) Suppl. 21, Circ. Mat. Palermo, Palermo, Italy, 1989, 15–52.
  • \lccD. B. Fuks [Fuchs], Cohomology of Infinite-Dimensional Lie Algebras, Contemp. Soviet Math., Consultants Bureau, New York, 1986.
  • \lccI. M. Gel'fand and D. A. Kazhdan, Some problems of differential geometry and the calculation of the cohomology of Lie algebras of vector fields, Sov. Math. Dokl. 12 (1971), 1367–1370.
  • \lccG. Hochschild and J.-P. Serre, Cohomology of Lie algebras, Ann. of Math. (2) 57 (1953), 591–603.
  • \lccM. Kontsevich, Quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), 157–216.
  • \lccS. Mac Lane, Homology, Classics Math., Springer, Berlin, 1975.
  • \lccR. Nest and B. Tsygan, Algebraic index theorem, Comm. Math. Phys. 172 (1995), 223–262.
  • ––––, “On the cohomology ring of an algebra” in Advances in Geometry, Progr. Math. 172, Birkhäuser, Boston, 1999, 337–370.
  • \lccB. Shoikhet, A proof of the Tsygan formality conjecture for chains, Adv. Math. 179 (2003), 7–37.
  • \lccB. Tsygan, “Formality conjectures for chains” in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc. Transl. Ser. 2 194, Amer. Math. Soc., Providence, 1999, 261–274.