Creating quantum projective spaces by deforming q-symmetric algebras

TL;DR

This paper constructs a broad class of quantum projective spaces as deformations of q-symmetric algebras, linking them to quadratic Poisson structures and providing explicit, convergent quantizations in line with Kontsevich's conjecture.

Contribution

It introduces a method to deform toric q-symmetric algebras into quantum projective spaces, establishing their relation to quadratic Poisson structures and explicit quantizations.

Findings

Constructed quantum projective spaces as Koszul, Calabi-Yau algebras.

Proved deformations are unobstructed under certain conditions.

Linked these algebras to explicit, convergent quantizations of quadratic Poisson structures.

Abstract

We construct a large collection of "quantum projective spaces", in the form of Koszul, Calabi-Yau algebras with the Hilbert series of a polynomial ring. We do so by starting with the toric ones (the q-symmetric algebras), and then deforming their relations using a diagrammatic calculus, proving unobstructedness of such deformations under suitable nondegeneracy conditions. We then prove that these algebras are identified with the canonical quantizations of corresponding families of quadratic Poisson structures, in the sense of Kontsevich. In this way, we obtain the first broad class of quadratic Poisson structures for which his quantization can be computed explicitly, and shown to converge, as he conjectured in 2001.