Symplectic microgeometry III: monoids

Abstract

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.

Symplectic microgeometry II: generating functions

“…Our first main result is that any symplectic micromorphism admits a global generating function. This is the best possible case when it comes to quantization via Fourier integral operators (as will be shown in a sequel [4]).…”

G-systems and deformation of G-actions on ${\mathbb {R}}^{d}$Rd

“…There are two main reasons to be interested in this family of deformations. The first can be found in the work of the first author, see [3], [4] and [5], where a new approach to quantization of Lagrangian submanifolds is proposed. The theory started in this note should represent the algebraic counterpart of the micro-symplectic approach to quantization developed in the cited papers.…”

Deformations of momentum maps and G-systems

“…Namely, the standard product defined by the composition of pseudo-differential operators as in (2.6) is in general not G-equivariant for cotangent lift actions (unless the action on R d we start with is linear). Thus, (C ∞ (T * R d )[[ ]], ⋆ st ) is in general not a quantum G-space in the sense of Definition (8), and Condition (1) is not satisfied. However, we have the following:…”

“…These symplectic micromorphisms always posses a global generating function (see [7]), which allows us to quantize them (i.e. to associate with them formal operators from C * (A)[[ ]] to C * (B)[[ ]], see [9]) using Fourier integral operator techniques. Formula (2.11) can be seen as such a quantization, where the symplectic micromorphism involved is the one that integrates the classical momentum map J, regarded as a Poisson map.…”