The finite-dimensional Groenewold–Van Hove conjecture

Let $M$ be a symplectic manifold and ${\cal B} \subset {\cal P}(M)$ a basic set with $\wp({\cal B})$ finite-dimensional. A strong quantization is understood in the sense used in the paper.

Groenewold–Van Hove conjecture. There is no nontrivial strong quantization of

$$\big({\cal P}(M),{\cal B}\big).$$

This conjecture proposes a general obstruction to strong quantization when the Poisson algebra generated by the basic set is finite-dimensional. The paper presents it as a speculative generalization based on examples, and its general status is not established there.