Nakayama automorphism

A noncommutative \(\mathbb{P}^3\) is in particular skew Calabi–Yau: there is a graded algebra automorphism \(\nu\) of \(A\), the Nakayama automorphism, and an integer \(\ell\) with a bimodule isomorphism

where \({}^{1}\!A^{\nu}\) is \(A\) as a bimodule with the right action twisted by \(\nu\). Equivalently, \(\nu\) is the twist that makes the superpotential cyclically invariant. The automorphism is the obstruction to \(A\) being Calabi–Yau:

It was put on a systematic footing — together with the homological identities it satisfies — by Reyes–Rogalski–Zhang.

Since \(A\) is connected graded and generated in degree \(1\), the Nakayama automorphism is graded, so it is determined by its action on \(A_1 = \langle x_1, x_2, x_3, x_4\rangle\): a single invertible \(4 \times 4\) matrix \(N\), with \(\nu(x_j) = \sum_i N_{ij}\,x_i\). Its determinant \(\det \nu\) — the homological determinant of the Nakayama automorphism — is a finer numerical invariant.

What happens across the catalogue

Computing \(\nu\) for every family (see below) turns up a clean picture.

• \(\nu\) is almost always a monomial matrix — a diagonal matrix or a signed permutation — the only exception being Jordan, whose Nakayama automorphism is unipotent.
• Calabi–Yau families. \(\nu = \mathrm{id}\) holds, for all parameter values, for the commutative ring, the Sklyanin algebra and its twist, Caines, generalized Clifford 1, Shelton–Tingey, the Ore extension of the commutative ring, and the central extension of Sklyanin.
• Is it constant over the parameter space? For the parametrised families the answer splits almost evenly. About a quarter have a Nakayama automorphism that does not move with the parameters — the Calabi–Yau ones above, and a second group with \(\nu = -\mathrm{id}\) (the three Goetz–Kirkman–Moore–Vashaw algebras, the \(\mathrm{S}_\infty\) pair, generalized Clifford 2, Clifford, the central-extension twist). For the rest — the double-Ore families, Vancliff, the deformed skew and skew families — \(\nu\) genuinely depends on the parameters.
• But the determinant never does. Even where \(\nu\) moves, it moves only by a torus scaling whose contribution cancels in the determinant: \(\det\nu\) is constant over the parameter space of every family, always landing in \(\{1, -1, i\}\) (it is \(-1\) for a handful of double-Ore families and generalized Clifford 3, and \(i\) for the \(\mathrm{A}_5\) family).

So the Nakayama automorphism is a genuinely moving invariant on these moduli, while its determinant is rigid. The determinant appears as its own column in the main table; the full automorphism is recorded on each family’s page.

Every family

The type of \(\nu\), whether the algebra is Calabi–Yau, whether \(\nu\) is constant across the parameters, and \(\det\nu\):

How it was computed

The Nakayama automorphism is computed from the superpotential with the AssociativeAlgebras package in Macaulay2 (superpotential, then nakayamaAut), extracting the \(4 \times 4\) matrix \(N\) on \(A_1\). The base field is chosen per family, and the constancy verdict is in every case cross-checked by computing \(\nu\) at two independent generic parameter points over \(\mathrm{GF}(60013)\) and comparing the matrices (a Nakayama automorphism that depended on the parameters would differ at two random points).

• Exactly over \(\mathbb{Q}(\text{params})\) for the families whose structure constants are rational — the closed forms on their pages (e.g. \(\operatorname{diag}(h^{-2}, h^{-2}, h^{2}, h^{2})\) for double Ore A) are genuine identities of rational functions, so “constant?” is read off directly.
• Exactly over \(\mathbb{F}_{60013}(\text{params})\) for the families whose definition uses a root of unity (\(i\) or a primitive cube root \(\omega\)) — the prime \(60013 \equiv 1 \bmod 12\) supplies both. (A genuine characteristic-zero computation would need \(\mathbb{Q}(i)\) or \(\mathbb{Q}(\omega)\), but the package requires its coefficient ring to be one of \(\mathbb{Z}/n\), \(\mathbb{Z}\), \(\mathbb{Q}\) or \(\mathrm{GF}\); the resulting matrix is field-independent, so this is the generic value.)
• At a generic point of \(\mathbb{F}_{60013}\) for the three families where a function-field Gröbner basis does not terminate — the central extension (\(12\) parameters), Clifford (\(24\)), and the Ore extension of the commutative ring — with constancy confirmed by sampling two independent points (the same method the rest of the site uses for many-parameter families).

Parameter-free families are computed over their field of definition (\(\mathbb{Q}\), or \(\mathbb{F}_{60013}\) when a root of unity appears). The reproduction script is code/nakayama.m2.

Per-invariant overview tables: point schemes, line schemes, \(\mathrm{HH}^i_0(A)\), \(\mathrm{HH}^i(\operatorname{qgr} A)\), the centre, normal elements, the Kodaira–Spencer map, and the Nakayama automorphism.