The centre
The centre \(\operatorname{Z}(A)\) of a graded algebra is the graded subalgebra of elements commuting with everything. For a generic noncommutative \(\mathbb{P}^3\) it is trivial in low degrees (\(\operatorname{Z}(A) = k\)); extra central elements signal extra geometry. The classic case is a pencil of central quadrics, as for the Sklyanin algebra.
Families with non-trivial centre
The dimensions \(\dim \operatorname{Z}(A)_n\) through degree 6, with the centre’s Hilbert series \(\operatorname{h}_{\operatorname{Z}(A)}(t) = \sum_n \dim_k \operatorname{Z}(A)_n \, t^n\):
| family | \(Z_1\) | \(Z_2\) | \(Z_3\) | \(Z_4\) | \(Z_5\) | \(Z_6\) | \(\operatorname{h}_{\operatorname{Z}(A)}(t)\) |
|---|---|---|---|---|---|---|---|
| Sklyanin, Sklyanin twist | 0 | 2 | 0 | 3 | 0 | 4 | \(1/(1-t^2)^2 = k[\Omega_1, \Omega_2]\) |
| Clifford | 0 | 4 | 0 | 10 | 0 | 20 | \(1/(1-t^2)^4 = k[x_1^2, \dots, x_4^2]\) |
| Generalized Clifford 2 | 0 | 2 | 0 | 5 | 0 | 8 | \(1 + 2t^2 + 5t^4 + 8t^6 + \cdots\) |
| \(\mathrm{S}_\infty\) twist | 0 | 1 | 0 | 2 | 0 | 2 | \(1 + t^2 + 2t^4 + 2t^6 + \cdots\) |
| central extension of Sklyanin | 1 | 1 | 2 | 2 | 2 | 3 | \(1 + t + t^2 + 2t^3 + 2t^4 + 2t^5 + 3t^6 + \cdots\) |
For Sklyanin and Clifford the centre is an honest polynomial ring — a pencil of central quadrics, respectively the four central squares. The Generalized Clifford 1 and Shelton–Tingey algebras have trivial centre in degree 2 but acquire central elements first in degree 4. (Generalized Clifford 3 is regular only at special parameters, where its centre depends on the chosen point.) Every other family has trivial centre in low degrees — the sole exception being the commutative ring, which is its own centre.
Every family
The number of independent central quadrics, cubics and quartics (\(\dim \operatorname{Z}(A)_2\), \(\dim \operatorname{Z}(A)_3\), \(\dim \operatorname{Z}(A)_4\)) for every family:
How it was computed
The degree-by-degree dimensions above were computed with Macaulay2 over a generic point of a finite field \(\mathrm{GF}(p)\) (\(p \equiv 1 \bmod 12\)), confirmed at two primes to pin down the generic value; an explicit noncommutative Gröbner basis to degree 8 ensures degree-\(\le 6\) central elements are not truncated. For the small-parameter families these agree with a computation over the function field \(\mathbb{Q}(\text{params})\); with many parameters (Clifford has 24, the central extension 12) a function-field Gröbner basis does not terminate, which is why the finite-field computation is used.