The Groenewold–Van Hove conjecture for finite-dimensional basic algebras

Let $M$ be a symplectic manifold with a finite-dimensional basic algebra ${\mathfrak b}$. Let $P({\mathfrak b})$ denote its polynomial algebra, and suppose that condition (D1), $1\in\{P({\mathfrak b}),P({\mathfrak b})\}$, or condition (D2), $P({\mathfrak b})$ is not free as an associative algebra, holds. Groenewold–Van Hove conjecture. There is no nontrivial quantization of $(P({\mathfrak b}),{\mathfrak b})$. The claim proposes a general form of the Groenewold–Van Hove obstruction: the two mechanisms that allow Poisson brackets to decrease polynomial degree should prevent quantization. The examples discussed support it, while the converse—that neither condition implies existence of a nontrivial quantization—is proved in the paper; the conjecture itself is presented without a resolution.