4d-AS-regular

the classification of 4-dimensional quadratic Artin–Schelter regular algebras

Families

The known families of four-dimensional Artin–Schelter regular algebras, on four generators \(x_1, x_2, x_3, x_4\).

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.

Deformations of the polynomial algebra

The commutative polynomial ring (set apart below), together with its quantum deformations as classified by Pym (arXiv:1403.6444): the skew polynomial ring, the Sklyanin algebra and its central extension, the Cassidy–Goetz–Shelton algebra, an Ore extension of a polynomial ring, and an exceptional component.

commutativeThe commutative polynomial ring in four variables: the generators commute. Its point scheme is all of \(\mathbb{P}^3\), its centre is the whole ring, and every element is normal — the commutative reference point against which the noncommutative families are measured.
SklyaninThe "elliptic" noncommutative \(\mathbb{P}^3\). The algebra is built from an elliptic curve \(E\) together with a translation automorphism \(σ\): the six quadratic relations encode theta-function identities on \(E\), and the point scheme is \(E\) itself (a quartic) together with four \(σ\)-fixed points. It carries a pencil of central quadrics \(k[\Omega_1, \Omega_2]\).
skewThe skew polynomial ring: each pair of generators \(q\)-commutes, \(x_i x_j = q_{ij}\, x_j x_i\). This is the most "abelian" noncommutative \(\mathbb{P}^3\), a toric deformation of the polynomial ring. Its point scheme is the \(1\)-skeleton of the coordinate tetrahedron — the six edges (a copy of \(K_4\)), two edges meeting iff they share a vertex.
central extension of SklyaninStart from a 3-dimensional regular algebra on \(x_1, x_2, x_3\) (here of Sklyanin/elliptic type) and adjoin a central generator \(x_4 = z\), deforming the three defining relations by terms linear in \(z\) plus a multiple of \(z^2\). The extension is again regular, of dimension 4, and is the generic way to build a noncommutative \(\mathbb{P}^3\) with a central hyperplane variable.
Cassidy–Goetz–SheltonThe Cassidy–Goetz–Shelton algebra: \(x_1, x_2\) commute, the pairs involving \(x_3, x_4\) \(q\)-commute, and the \(x_3 x_4\) relation carries a quadratic correction in \(x_1, x_2\). The \(σ\) is a diagonal twist (the parameters \(α, β\)), under which the family is closed; the \(σ\)-twisted form is the one appearing in the dimension-four classification.
Ore extension of commutativeA quantization of a Poisson structure on \(\mathbb{P}^3\) whose degeneracy divisor is a configuration of three planes and a cubic. The variables \(x_1, x_2, x_3\) commute among themselves, while the fourth variable \(x_4\) acts on them through a derivation built from the cyclic data \((c_i, d_i)\). The point scheme is a union of three lines and a rational normal quartic.
JordanAn "additive" deformation: the commutators of the generators are prescribed quadratic forms with coefficients polynomial in the single parameter \(a\), realising a quantization of a Poisson structure whose semiclassical limit has a \(\mathbb{G}_{\mathrm{a}}\)-symmetry. Its point modules are parametrised by a bouquet of rational normal curves (a line, a conic and a twisted cubic). It is called *Jordan* because it is related to the Jordan plane \(k\langle x, y\rangle/(yx - xy - x^2)\): quotienting repeatedly by the first variable \(x_1\) recovers the Jordan plane.

Named families

VancliffBuilt so that its point scheme is the union of a quadric surface and a line in \(\mathbb{P}^3\) — the first examples of regular algebras whose point scheme is \(2\)-dimensional. Five of the six relations are skew-commutation (\(q\)-commuting) relations; the last couples \(x_3 x_2\) to \(x_1 x_4\).
Vancliff twistObtained from the Vancliff algebra by changing the sign in the relations involving \(x_4\) (so two \(q\)-commutators become \(q\)-anticommutators). The point scheme degenerates to a configuration of five lines.
CliffordEach pair of generators anticommutes up to a quadratic correction that is a linear combination of the squares \(x_ℓ^2\). Equivalently the algebra is determined by a symmetric \(4 \times 4\) matrix of quadratic forms; it is Artin–Schelter regular exactly when the associated system of quadrics is base-point free. The four squares span a \(4\)-dimensional space of central quadrics.
Shelton–TingeyA rigid (parameter-free) Koszul AS-regular algebra obtained from Shelton–Tingey's construction; the squares \(x_1^2, x_2^2, x_3^2, x_4^2\) are paired and the cross relations are twisted by \(i\).
CainesA Koszul AS-regular algebra on four generators in which \(x_1, x_2\) anticommute and \(x_4\) acts on \(x_1, x_2, x_3\) through relations mixing the squares \(x_3^2\) with the products \(x_i x_4\). Its point scheme is a finite set of 13 points and it carries a pencil of central quadrics.
Generalized Clifford 1A graded skew Clifford algebra: the symmetric Clifford relations are twisted by roots of unity, so \(x_4, x_1\) and \(x_3, x_2\) skew-commute while the remaining relations carry quadratic corrections. Such algebras are regular when the defining quadrics are normalizing and base-point free.
Generalized Clifford 2Another graded skew Clifford algebra, presented through symmetric and skew-symmetric relations whose quadratic corrections are multiples of the squares \(x_2^2, x_3^2\). It carries a \(2\)-dimensional space of central quadrics.
Generalized Clifford 3A graded skew Clifford algebra in which three pairs \(q\)-commute and the remaining relations couple the squares \(x_1^2, x_4^2\) to the products \(x_2 x_3, x_2 x_4\). Regularity holds only on a constraint subvariety of the parameters.
\(\mathrm{S}_{d,i}\)Take the six defining relations \(f_1, \dots, f_6\) of the Sklyanin algebra and, for one chosen index \(i \in \{1, \dots, 6\}\), replace \(f_i\) by the single quadric \(d_1 \Omega_1 + d_2 \Omega_2\), where \(\Omega_1, \Omega_2\) are the two central quadrics of the Sklyanin algebra and \((d_1 : d_2) \in \mathbb{P}^1\). The five remaining Sklyanin relations \(f_j\) (\(j \neq i\)) are kept verbatim. As \(d \to \infty\) (only the quadrics surviving) one recovers \(\mathrm{S}_\infty\).
\(\mathrm{S}_{d,i}\) twistAs for \(\mathrm{S}_{d,i}\), but starting from the twisted Sklyanin relations \(f'_1, \dots, f'_6\) (commutators exchanged with anticommutators): for one chosen index \(i \in \{1, \dots, 6\}\), replace \(f'_i\) by the single twisted quadric \(d_1 \Omega'_1 + d_2 \Omega'_2\), keeping the five remaining \(f'_j\) (\(j \neq i\)). The signs in the quadrics change as for the \(\mathrm{S}_\infty\) twist.
\(\mathrm{S}_\infty\)\(\mathrm{S}_\infty\) keeps the first four (commutator/anticommutator) Sklyanin relations but replaces the last two by the quadrics \(\Omega_1 = -x_1^2 + x_2^2 + x_3^2 + x_4^2\) and \(\Omega_2 = x_2^2 + \tfrac{1+α}{1-β} x_3^2 + \tfrac{1-α}{1+γ} x_4^2\), imposed as relations. This kills the central quadrics yet keeps the elliptic quartic as point scheme.
\(\mathrm{S}_\infty\) twistAs for \(\mathrm{S}_\infty\), but the surviving Sklyanin relations are twisted (commutators exchanged with anticommutators) and the signs in the two quadrics change. The twist trades the elliptic point scheme for 20 points and produces a single central quadric whose symbol is the smooth hyperbolic form \(\mathbb{P}^1 \times \mathbb{P}^1\).
Sklyanin twistA 2-cocycle (comodule) twist preserves Hilbert series and AS-regularity but changes the geometry: the elliptic point scheme of the Sklyanin algebra is replaced by exactly 20 point modules together with infinitely many fat-point modules of multiplicity \(2\) (Davies). It is an "exotic elliptic algebra" in the sense of Chirvasitu–Smith. The pencil of central quadrics is preserved.
Goetz–Kirkman–Moore–Vashaw RA monomial-type quadratic algebra with no scalar parameters, built from the combinatorics of a dual reflection group. Each relation pairs two distinct monomials with sign \(\pm 1\); the Kodaira–Spencer map vanishes, so the family is a rigid point of the moduli space.
Goetz–Kirkman–Moore–Vashaw SA quadratic algebra with no scalar parameters coming from a dual reflection group: four of the relations identify a pair of mixed monomials, while two relations equate a product with a square \(x_3^2\) or \(x_4^2\). The Kodaira–Spencer map vanishes, so the family is a rigid point of the moduli space.
Goetz–Kirkman–Moore–Vashaw TA quadratic algebra with no scalar parameters coming from a dual reflection group, a sign variant of the algebra "S" above. The Kodaira–Spencer map vanishes, so the family is a rigid point of the moduli space.
\(\mathrm{A}_5\)A four-parameter quadratic algebra defined over a field containing \(\sqrt{-1}\). Three relations make \(x_4\) skew/normalize against \(x_1, x_2, x_3\) (with the coefficient \(i\) appearing in the \(x_3, x_4\) relation), and two further relations impose quadratic identities among the squares \(x_1^2, x_2^2, x_3^2\).
central extension of Sklyanin twistObtained by applying a 2-cocycle (Klein four-group) twist to a central extension of a 3-dimensional regular algebra. The variable \(x_3\) plays the role of the adjoined central-type generator: it commutes with each of \(x_1, x_2, x_4\), while the first three relations \(q\)-commute the remaining pairs and absorb quadratic corrections \(b x_\bullet^2 + c_\bullet x_3^2\).
deformed skew \((x_3x_2;\ x_4^2)\)A quantum (skew) polynomial ring with a single quadratic term added: the \(x_3 x_2\) relation acquires \(-t\,x_4^2\).
deformed skew \((x_2x_1;\ x_3x_4)\)A quantum (skew) polynomial ring with a single quadratic term added: the \(x_2 x_1\) relation acquires the cross-term \(-t\,x_3 x_4\).
deformed skew \((x_3x_2;\ x_4^2, x_1^2)\)A quantum (skew) polynomial ring with two quadratic terms added to one relation: \(x_3 x_2\) acquires \(-t\,x_4^2 - s\,x_1^2\).
deformed skew \((x_3x_1, x_3x_2;\ x_4^2)\)A quantum (skew) polynomial ring deformed in two relations by \(x_4^2\): the \(x_3 x_1\) relation acquires \(-t\,x_4^2\) and the \(x_3 x_2\) relation \(-s\,x_4^2\).
deformed skew \((x_2x_1, x_3x_2;\ x_3^2, x_4^2)\)A quantum (skew) polynomial ring deformed in two relations: the \(x_2 x_1\) relation acquires \(-t\,x_3^2\) and the \(x_3 x_2\) relation \(-s\,x_4^2\).

Double Ore extensions

The 26 double Ore extensions of Zhang–Zhang, labelled A–Z.