Formalite $G_\infty$ adaptee et star-representations sur des sous-varietes coisotropes

TL;DR

This paper generalizes Tamarkin's formality to construct star products on Poisson manifolds with coisotropic submanifolds, enabling representations that deform classical function representations, especially in flat cases.

Contribution

It extends Tamarkin's formality to coisotropic submanifolds, providing a method to construct star products with specific ideal properties.

Findings

No obstructions to formality in flat cases like R^n with R^{n-l}

Constructs star products making the ideal I a left ideal

Provides a representation of the deformed algebra on quotient spaces

Abstract

Let X be a Poisson manifold and C a coisotropic submanifold and let I be the vanishing ideal of C. In this work we want to construct a star product * on X such that I[[lambda]] is a left ideal for *. Thus we obtain a representation of the star product algebra A[[lambda]] = C^\infty(X)[[lambda]] on B[[lambda]] = A[[lambda]] / I[[lambda]] deforming the usual representation of A on the functions on C. The result follows from a generalization of Tamarkin's formality adapted to the submanifold C. We show that in the case X = R^n and C = R^{n-l} with l > 1 there are no obstructions to this formality.