Obstructions for Twist Star Products

TL;DR

This paper demonstrates that not all star products originate from Drinfel'd twists, highlighting that some Poisson structures, such as those on certain surfaces, cannot be induced by classical r-matrices.

Contribution

It provides counterexamples showing that certain Poisson structures are not induced by classical r-matrices, challenging assumptions about the universality of twist-induced star products.

Findings

Not all Poisson structures are induced by classical r-matrices.

Examples include higher genus symplectic Pretzel surfaces.

The symplectic sphere $\

Abstract

In this short note we point out that not every star product is induced by a Drinfel'd twist by showing that not every Poisson structure is induced by a classical $r$-matrix. Examples include the higher genus symplectic Pretzel surfaces and the symplectic sphere $\mathbb{S}^2$.