# Formal symplectic groupoid

Cattaneo, A S; Dherin, B; Felder, G (2005). Formal symplectic groupoid. Communications in Mathematical Physics, 253(3):645-674.

## Abstract

The multiplicative structure of the trivial symplectic groupoid over ℝᵈ 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 Kontsevichs 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 ODEs.

## Abstract

The multiplicative structure of the trivial symplectic groupoid over ℝᵈ 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 Kontsevichs 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 ODEs.

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## Additional indexing

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2005 27 Jan 2010 12:40 19 Sep 2018 10:37 Springer 0010-3616 The original publication is available at www.springerlink.com Green https://doi.org/10.1007/s00220-004-1199-z

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