The Universal Generating Function of Analytical Poisson Structures

The notion of generating functions of Poisson structures was first studied in math.SG/0312380.They are special functions which induce, on open subsets of $\R^d$, a Poisson structure together with the local symplectic groupoid integrating it. A universal generating function was provided in terms of a formal power series coming from Kontsevich star product. The present article proves that this universal generating function converges for analytical Poisson structures and compares the induced local symplectic groupoid with the phase space of Karasev--Maslov.Comment: 15 pages, 2 figures, shorter version, introductive part remove

Deformation quantization of Leibniz algebras

This paper has two parts. The first part is a review and extension of the methods of integration of Leibniz algebras into Lie racks, including as new feature a new way of integrating 2-cocycles (see Lemma 3.9). In the second part, we use the local integration of a Leibniz algebra h using a Baker-Campbell-Hausdorff type formula in order to deformation quantize its linear dual h^*. More precisely, we define a natural rack product on the set of exponential functions which extends to a rack action on C^{\infty}(h^*).Comment: 37 pages, added explicit computation of the first term of the deformation, i.e. the bracket which is to be quantize

The Universal Generating Function of Analytical Poisson Structures

Generating functions of Poisson structures are special functions which induce, on open subsets of $$\mathbb{R}^d$$ , a Poisson structure together with the local symplectic groupoid integrating it. In a previous paper by A. S. Cattaneo, G. Felder and the author, a universal generating function was provided in terms of a formal power series coming from Kontsevich star product. The present article proves that this universal generating function converges for analytical Poisson structures and shows that the induced local symplectic groupoid coincides with the phase space of Karasev-Maslo

Formal symplectic realizations

We study the relationship between several constructions of symplectic realizations of a given Poisson manifold. Our main result is a general formula for a formal symplectic realization in the case of an arbitrary Poisson structure on $\R^n$. This formula is expressed in terms of rooted trees and elementary differentials, building on the work of Butcher, and the coefficients are shown to be a generalization of Bernoulli numbers appearing in the linear Poisson case. We also show that this realization coincides with a formal version of the original construction of Weinstein, when suitably put in global Darboux form, and with the realization coming from tree-level part of Kontsevich's star product. We provide a simple iterated integral expression for the relevant coefficients and show that they coincide with underlying Kontsevich weights.Comment: 21 pages, 4 figures. Revised version. Published in IMR

Symplectic Microgeometry III: Monoids

We show that the category of Poisson manifolds and Poisson maps, the category of symplectic microgroupoids and lagrangian submicrogroupoids (as morphisms), and the category of monoids and monoid morphisms in the microsymplectic category are equivalent symmetric monoidal categories.Comment: 19 pages, 2 figure

Integration of Lie Algebroid Comorphisms

We show that the path construction integration of Lie algebroids by Lie groupoids is an actual equivalence from the category of integrable Lie algebroids and complete Lie algebroid comorphisms to the category of source 1-connected Lie groupoids and Lie groupoid comorphisms. This allows us to construct an actual symplectization functor in Poisson geometry. We include examples to show that the integrability of comorphisms and Poisson maps may not hold in the absence of a completeness assumption.Comment: 28 pages, references adde

Tensor products of representations up to homotopy

We study the construction of tensor products of representations up to homotopy, which are the A-infinity version of ordinary representations. We provide formulas for the construction of tensor products of representations up to homotopy and of morphisms between them, and show that these formulas give the homotopy category a monoidal structure which is uniquely defined up to equivalence.Comment: 42 pages, 2 figure