zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Formal symplectic groupoid. (English) Zbl 1072.58008

The principal objective in this paper is to give a positive answer to the deformation problem for symplectic groupoids. Its exact formulation goes as follows:

Given a Poisson structure α on d there exists a unique natural deformation of the trivial generating function such that the first-order is precisely α. Moreover, we have an explicit formula for this deformation

S h (p 1 ,p 2 ,x)=x(p 1 +p 2 )+ n=1 h n n! ΓT n,2 W Γ B ^ Γ (p 1 ,p 2 ,x),

where T n,2 is the set of Kontsevich trees of type (n,2), W Γ is the Kontsevich weight of Γ and B ^ Γ is the symbol of the bidifferential operator B Γ associated to Γ.

In case of a linear Poisson structure (i.e., the Kirillov-Kostant Poisson structure on the dual 𝒢 * of a Lie algebra 𝒢), the generating function of the sympletic groupoid over 𝒢 * reduces exactly to the familiar Campbell-Baker-Hausdorff formula

S h (p 1 ,p 2 ,x)=1 h C B H (hp 1 ,hp 2 ) , x,

where , is the natural pairing between 𝒢 and 𝒢 * .

Finally, the authors show that the star-product studied in [M. Kontsevich, Lett. Math. Phys. 66, No. 3, 167–216 (2003; Zbl 1058.53065)] can be considered as a suitable exponentiation of a deformation of the Poisson structure.

58H05Pseudogroups and differentiable groupoids on manifolds
53D05Symplectic manifolds, general
[1]Bates, S., Weinstein, A.: Lectures on the geometry of quantization. Berkeley Mathematics Lecture Notes, 8. Providence, RI: and Berkeley, CA: American Mathematical Society, Berkeley Center for Pure and Applied Mathematics, 1997
[2]Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. I. Deformations of symplectic structures. Ann. Physics 111(1), 61-110 (1978)
[3]Cattaneo, A.S.: The Lagrangian operad. Unpublished notes, http://www.math.unizh.ch/asc/lagop.pdf
[4]Cattaneo, A.S., Felder, G.: Poisson sigma models and deformation quantization. Euroconference on Brane New World and Noncommutative Geometry (Torino, 2000). Modern Phys. Lett. A 16(4-6), 179-189 (2001)
[5]Cattaneo, A.S., Felder, G.: Poisson sigma models and symplectic groupoids. In: Quantization of singular symplectic quotients, Progr. Math. 198, Basel: Birkhäuser, 2001, pp. 61-93
[6]Coste, A., Dazord, P., Weinstein, A.: Groupoïdes symplectiques. (French) [Symplectic groupoids] Publications du Département de Mathématiques. Nouvelle Série. A, Vol. 2, i?ii, Publ. Dép. Math. Nouvelle Sér. A, 87-2, Lyon: Univ. Claude-Bernard, 1987, pp. 1-62
[7]Crainic, M.: Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes. http://arxiv.org/math.DG/0008064, 2000
[8]Crainic, M., Fernandes, R.L.: Integrability of Lie brackets. Ann. of Math. (2) 157(2), 575-620 (2003)
[9]Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. In: Structure-preserving algorithms for ordinary differential equations. Springer Series in Computational Mathematics, 31, Berlin: Springer-Verlag, 2002
[10]Karabegov, K.: On Dequantization of Fedosov?s Deformation Quantization. http://arxiv.org/abs/math.QA/0307381, 2003
[11]Karasëv, M.V.: Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 50(3), 508-538, 638 (1986)
[12]Kathotia, V.: Kontsevich?s universal formula for deformation quantization and the Campbell-Baker-Hausdorff formula. Internat. J. Math. 11(4), 523-551 (2000)
[13]Kontsevich, M.: Deformation quantization of Poisson manifolds, I. Lett. Math. Phys. 66, 157-216 (2003) · Zbl 1058.53065 · doi:10.1023/B:MATH.0000027508.00421.bf
[14]Weinstein, A.: Symplectic groupoids and Poisson manifolds. Bull. Am. Math. Soc. (N.S.) 16(1), 101-104 (1987) · Zbl 0618.58020 · doi:10.1090/S0273-0979-1987-15473-5
[15]Weinstein, A., Xu, P.: Extensions of symplectic groupoids and quantization. J. Reine Angew. Math. 417, 159-189 (1991)
[16]Weinstein, A.: Noncommutative geometry and geometric quantization. In: Symplectic geometry and mathematical physics (Aix-en-Provence, 1990), Progr. Math. 99, Boston, MA: Birkhäuser Boston, 1991, pp. 446-461
[17]Weinstein, A.: Tangential deformation quantization and polarized symplectic groupoids. In: Deformation theory and symplectic geometry (Ascona, 1996), Math. Phys. Stud. 20, Dordrecht: Kluwer Acad. Publ. 1997, 301-314
[18]Zakrzewski, S.: Quantum and classical pseudogroups. I. Union pseudogroups and their quantization. Commun. Math. Phys. 134(2), 347-370 (1990)