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Communications in Mathematical Physics

, Volume 253, Issue 3, pp 645–674 | Cite as

Formal Symplectic Groupoid

  • Alberto S. CattaneoEmail author
  • Benoit Dherin
  • Giovanni Felder
Article

Abstract

The multiplicative structure of the trivial symplectic groupoid over ℝ d associated to the zero Poisson structure can be expressed in terms of a generating function. We address the problem of deforming such a generating function in the direction of a non-trivial Poisson structure so that the multiplication remains associative. We prove that such a deformation is unique under some reasonable conditions and we give the explicit formula for it. This formula turns out to be the semi-classical approximation of Kontsevich’s deformation formula. For the case of a linear Poisson structure, the deformed generating function reduces exactly to the CBH formula of the associated Lie algebra. The methods used to prove existence are interesting in their own right as they come from an at first sight unrelated domain of mathematics: the Runge–Kutta theory of the numeric integration of ODE’s.

Keywords

Neural Network Statistical Physic Generate Function Complex System Nonlinear Dynamics 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Alberto S. Cattaneo
    • 1
    Email author
  • Benoit Dherin
    • 2
  • Giovanni Felder
    • 2
  1. 1.Institut für MathematikUniversität Zürich–IrchelZürichSwitzerland
  2. 2.D-MATHETH-ZentrumZürichSwitzerland

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