Formal Symplectic Groupoid

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.

The Universal Generating Function of Analytical Poisson Structures

“…This is the main result of [2]. This universal generating function turns out to be the semi-classical part of Kontsevich star product associated to α on U ,…”

“…Thus, G U (S) inherits the canonical symplectic structure of T * U , turning G U (S) into a symplectic manifold. We proved in [2] and [5] that the matrix…”

“…In [2] and in [5], it was shown that if S satisfies the SGS conditions and the SGA equation 1 (see Section 2), then S generates a Poisson structure on U given by α(x) := ∂ 2 S ∂p 1 ∂p 2 (0, 0, x)…”

“…The results presented here are part of the author PhD thesis [5] and may be seen as a natural development of [2].…”

“…The notion of generating functions of Poisson structures was first studied in [2]. They are special functions which induce, on open subsets of R d , a Poisson structure together with the local symplectic groupoid integrating it.…”