Normal elements

A homogeneous element \(n\) of a graded algebra \(A\) is normal if \(nA = An\) — the two-sided analogue of a central element (every central element is normal). Normal elements of degree \(1\) cut out hyperplanes that are again “coordinate” subalgebras, and degree-\(2\) normal elements include the central quadrics; they control much of the ring-theoretic geometry.

Normality is a determinantal condition, so the normal elements of a fixed degree \(d\) form a Zariski-closed cone in \(A_d\). They do not in general form a linear subspace: a sum of two normal elements need not be normal, and our computation (the rank of a parameter-dependent map) determines only the dimension of this locus — it does not establish linearity. The table gives the projective dimension of the normal locus in \(\mathbb{P}(A_d)\) at the generic point, for \(d = 1, 2\):

• \(-1\): no normal element in that degree;
• \(0\): a finite set of normal directions;
• \(1, 3, \dots\): a positive-dimensional family of normal elements.

When the normal locus happens to coincide with the (linear) space of central elements it is a linear subspace — for Sklyanin the degree-2 locus is the central pencil \(\langle \Omega_1, \Omega_2 \rangle\), and for Clifford it is the \(\mathbb{P}^3\) of central squares — but this is special to those families.

Computed at generic parameters over \(\mathrm{GF}(p)\) (\(p \equiv 1 \bmod 12\)), confirmed at two primes. The validation \(\dim_{\mathbb{P}}(\text{normal}_2) \ge \dim \operatorname{Z}(A)_2 - 1\) (central ⊆ normal) holds throughout. Generalized Clifford 3 is omitted: it is regular only on its constraint variety, where a generic parameter point does not land.