Constructing and classifying quantizations of projective space for n ≥ 5

Determine a complete classification and explicit construction of homogeneous quantizations of the homogeneous coordinate ring C[x_0, ..., x_{n-1}] of projective space P^{n−1} for n ≥ 5, i.e., associative flat deformations defined by quadratic relations that deform the commutation relations x_i x_j = x_j x_i while preserving the expected homological properties (Koszul, Artin–Schelter regular, Calabi–Yau), up to the action of GL(n).

Background

The paper studies non-commutative, associative deformations (quantizations) of polynomial algebras associated to projective spaces. For P^{n−1}, quantizations correspond to quadratic deformations of the commutation relations in the homogeneous coordinate ring. While complete classifications exist for low dimensions (n ≤ 3) and there is substantial progress for n = 4 via Poisson bracket classifications, the situation for n ≥ 5 remains largely unresolved.

The authors construct a large family of explicit analytic quantizations for odd n ≥ 3 under genericity conditions, providing lower bounds on components of the moduli space (e.g., at least 40 components for n = 5). Despite these advances, a full classification of all such quantizations for n ≥ 5 remains an open challenge.