, Volume 253, Issue 3, pp 645-674

Formal Symplectic Groupoid

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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.

Communicated by L. Takhtajan
A.S.C. acknowledges partial support of SNF Grant No. 20-100029/1.
B.D. and G.F. acknowledge partial support of SNF Grant No. 21-65213.01.
Acknowledgement The second author thanks Ernst Hairer for useful discussions, and suggestions.