# All family data for the 4d-AS-regular website.
# Assembled from data/families/*.yaml; one entry per family, keyed by slug.

a5:
  name: '\(\mathrm{A}_5\)'
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 4
  hh0: [2, 1, 0]
  hh_qgr: [1, 1, 1, 5]
  line_scheme_dim: 2
  line_scheme: 'a plane and a curve of degree 11'
  year: 2025
  slug: "a5"
  sortkey: "2025 a5"
  ks_rank: 1
  ks_inj: false
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  parameters:
    - symbol: "d, a_1, a_4, a_7"
      description: >
        Four scalars in the base field. The relations also involve the primitive
        fourth root of unity \(i\) (with \(i^2 = -1\)).
  introduced:
    reference: "2511.08390"
    note: >
      \(\mathrm{A}_5\) appears among the families mapping densely onto irreducible
      components of the moduli space of regular algebras (its Kodaira–Spencer map
      is surjective of rank 1). No earlier bibliographic source is identified for
      it.
  construction: >
    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 \(x_1^2, x_2^2, x_3^2\).
  relations:
    - "d x_1 x_4 + x_4 x_1"
    - "d x_2 x_4 - x_4 x_2"
    - "a_1 x_1^2 + a_4 d^2 x_2^2"
    - "d x_3 x_4 - i x_4 x_3"
    - "a_4 x_1 x_2 + a_4 x_2 x_1 - a_7 x_3^2"
    - "x_2 x_3 + x_3 x_2"
  references:
    - "2511.08390"
  notes: >
    \(\mathrm{A}_5\) is defined over a field containing \(\sqrt{-1}\) (here
    \(\mathbb{Q}(i)\), with \(i\) the primitive fourth root of unity appearing in
    the relations); its invariants are computed over that field.

  point_scheme_dim: 1
  point_scheme: 'a conic and a cubic curve (degree 5)'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}(-d^{-1},d^{-1},i\,d^{-1},-d^{3})'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: "i"
  nakayama_field: 'gf-params'
caines:
  name: "Caines"
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 4
  hh0: [1, 8, 21]
  hh_qgr: [1, 0, 14, 19]
  point_scheme_dim: 0
  line_scheme_dim: 1
  line_scheme: 'a degree-17 curve, a conic and a line'
  point_scheme: '13 points'
  year: 2005
  slug: "caines"
  sortkey: "2005 caines"
  ks_rank: 1
  ks_inj: false
  ks_surj: false
  centre_z2: 2
  centre_z3: 0
  centre_z4: 3
  parameters:
    - symbol: "a, b, c, d"
      description: "Four scalars (\\(c \\neq 0\\))."
  introduced:
    reference: "caines-thesis"
    note: >
      Introduced in Ian Caines's PhD thesis, "New examples of four-dimensional
      Artin–Schelter regular algebras"; it appears as the Caines algebra in the
      dimension-4 classification.
  construction: >
    A 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.
  relations:
    - "x_4 x_3 - x_3 x_4 - a x_1 x_2"
    - "x_4 x_2 - b x_3^2 + x_2 x_4"
    - "x_4 x_1 - c x_3^2 + x_1 x_4"
    - "x_3 x_2 - x_2 x_3 + (b d / c) x_2 x_4 - (b^2 d / c^2) x_1 x_4"
    - "x_3 x_1 - x_1 x_3 + (b d / c) x_1 x_4 - d x_2 x_4"
    - "x_2 x_1 + x_1 x_2"
  references:
    - "caines-thesis"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: true
  nakayama: '\mathrm{id}'
  nakayama_type: 'identity'
  nakayama_constant: true
  homological_det: 1
  nakayama_field: 'qq'
cassidy-goetz-shelton:
  name: "Cassidy–Goetz–Shelton"
  kind: "named"
  pym: true
  poisson: true
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 5
  hh0: [2, 4, 8]
  hh_qgr: [1, 1, 3, 7]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'two planes, two conics and two lines'
  year: 2006
  slug: "cassidy-goetz-shelton"
  sortkey: "2006 cassidy-goetz-shelton"
  ks_rank: 4
  ks_inj: false
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "p_0, p_1, λ, α, β"
      description: >
        \(p_0, p_1\) nonzero scalars; \(λ, α, β\) scalars (\(α, β\) rescaling the
        \(σ\)-twist).
  introduced:
    reference: "MR2253666"
    note: >
      A generalized Laurent polynomial ring of Cassidy–Goetz–Shelton; in Pym's
      classification of quadratic Poisson structures it is the type \(L(1,1,2)\).
  construction: >
    The 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.
  relations:
    - "α (x_2 x_1 - x_1 x_2)"
    - "β x_3 x_1 - (1/p_0) α x_1 x_3"
    - "(α^2/β) x_4 x_1 - p_0 α x_1 x_4"
    - "β x_3 x_2 - p_1 α x_2 x_3"
    - "(α^2/β) x_4 x_2 - (1/p_1) α x_2 x_4"
    - "(α^2/β) x_4 x_3 - p_1 (1/p_0) β x_3 x_4 - α (p_1 - p_0)(x_1^2 + λ x_1 x_2 + x_2^2) - α (1 - p_0^2) x_1^2 - α (p_1^2 - 1) x_2^2"
  references:
    - "MR2253666"
    - "MR3366864"
  point_scheme: 'a line, two conics and two points'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(1,1,\tfrac{a^{4}}{b^{4}},\tfrac{b^{4}}{a^{4}}\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
central-extension:
  name: "central extension of Sklyanin"
  kind: "named"
  pym: true
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 12
  hh0: [1, 7, 19]
  hh_qgr: [1, 0, 7, 12]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a plane and a curve of degree 15'
  year: 1996
  slug: "central-extension"
  sortkey: "1996 central-extension"
  ks_rank: 7
  ks_inj: false
  ks_surj: true
  centre_z2: 1
  centre_z3: 2
  centre_z4: 2
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "a, b, c"
      description: "Coefficients of the three base relations."
    - symbol: "α_1, α_2, α_3, l_{ij}"
      description: >
        The data of a central element: a symmetric \(3 \times 3\) matrix
        \((l_{ij})\) and a vector \((α_i)\), describing how the central generator
        \(x_4\) enters.
  introduced:
    reference: "MR1429334"
    note: >
      A central extension of a 3-dimensional Artin–Schelter regular algebra, in
      the sense of Le Bruyn–Smith–Van den Bergh.
  construction: >
    Start 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.
  relations:
    - "c x_1^2 + a x_2 x_3 + b x_3 x_2 + l_{11} x_1 x_4 + l_{12} x_2 x_4 + l_{13} x_3 x_4 + α_1 x_4^2"
    - "c x_2^2 + a x_3 x_1 + b x_1 x_3 + l_{12} x_1 x_4 + l_{22} x_2 x_4 + l_{23} x_3 x_4 + α_2 x_4^2"
    - "c x_3^2 + a x_1 x_2 + b x_2 x_1 + l_{13} x_1 x_4 + l_{23} x_2 x_4 + l_{33} x_3 x_4 + α_3 x_4^2"
    - "x_1 x_4 - x_4 x_1"
    - "x_2 x_4 - x_4 x_2"
    - "x_3 x_4 - x_4 x_3"
  references:
    - "MR1429334"
  point_scheme: 'a twisted cubic and 8 points'
  centre: '\(\dim \operatorname{Z}(A)_2 = 1\), \(\dim \operatorname{Z}(A)_3 = 2\) (a central quadric and central cubics by construction); \(x_4\) is central'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: true
  nakayama: '\mathrm{id}'
  nakayama_type: 'identity'
  nakayama_constant: true
  homological_det: 1
  nakayama_field: 'gf-sample'
central-extension-twist:
  name: "central extension of Sklyanin twist"
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 5
  hh0: [1, 4, 9]
  hh_qgr: [1, 0, 7, 12]
  line_scheme_dim: 2
  line_scheme: 'a plane and a curve of degree 15'
  year: 2025
  slug: "central-extension-twist"
  sortkey: "2025 central-extension-twist"
  ks_rank: 4
  ks_inj: false
  ks_surj: true
  centre_z2: 1
  centre_z3: 0
  centre_z4: 1
  parameters:
    - symbol: "q, b, c_0, c_1, c_2"
      description: "Five scalar parameters of the twisted central extension."
  introduced:
    reference: "MR1429334"
    note: >
      A cocycle twist of the central extension of a 3-dimensional Sklyanin-type
      Artin–Schelter regular algebra (Le Bruyn–Smith–Van den Bergh). It appears as
      "Central Extensions twist" in Tables 7 and 8 of arXiv:2511.08390, attributed
      there to Remark 5.13, and is the cocycle twist of the central-extension
      family in the same spirit that the Sklyanin twist relates to the Sklyanin
      algebra.
  construction: >
    Obtained 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 \(b x_\bullet^2 + c_\bullet x_3^2\).
  relations:
    - "x_2 x_1 + q x_1 x_2 + b x_4^2 + c_0 x_3^2"
    - "x_4 x_2 + q x_2 x_4 + b x_1^2 + c_1 x_3^2"
    - "x_1 x_4 + q x_4 x_1 + b x_2^2 + c_2 x_3^2"
    - "x_4 x_3 + x_3 x_4"
    - "x_1 x_3 + x_3 x_1"
    - "x_2 x_3 + x_3 x_2"
  references:
    - "MR1429334"

  point_scheme_dim: 1
  point_scheme: 'a cubic curve and 8 points'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '-\mathrm{id}'
  nakayama_type: 'scalar'
  nakayama_constant: true
  homological_det: 1
  nakayama_field: 'qq'
clifford:
  name: "Clifford"
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 24
  hh0: [1, 9, 19]
  hh_qgr: [1, 0, 15, 20]
  point_scheme_dim: 0
  line_scheme_dim: 2
  line_scheme: 'a surface of degree 10'
  year: 1995
  slug: "clifford"
  sortkey: "1995 clifford"
  ks_rank: 9
  ks_inj: false
  ks_surj: true
  centre_z2: 4
  centre_z3: 0
  centre_z4: 10
  normal_1: -1
  normal_2: 3
  parameters:
    - symbol: "a_{ijℓ}"
      description: >
        For each pair \(i < j\), the four coefficients of \(x_1^2, x_2^2, x_3^2,
        x_4^2\) in the corresponding relation — equivalently four symmetric
        matrices of quadrics (24 scalars).
  introduced:
    reference: "MR1356364"
    note: >
      A graded Clifford algebra; the graded skew generalisation is due to
      Cassidy–Vancliff.
  construction: >
    Each 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.
  relations:
    - "x_1 x_2 + x_2 x_1 + a_{121} x_1^2 + a_{122} x_2^2 + a_{123} x_3^2 + a_{124} x_4^2"
    - "x_1 x_3 + x_3 x_1 + a_{131} x_1^2 + a_{132} x_2^2 + a_{133} x_3^2 + a_{134} x_4^2"
    - "x_1 x_4 + x_4 x_1 + a_{141} x_1^2 + a_{142} x_2^2 + a_{143} x_3^2 + a_{144} x_4^2"
    - "x_2 x_3 + x_3 x_2 + a_{231} x_1^2 + a_{232} x_2^2 + a_{233} x_3^2 + a_{234} x_4^2"
    - "x_2 x_4 + x_4 x_2 + a_{241} x_1^2 + a_{242} x_2^2 + a_{243} x_3^2 + a_{244} x_4^2"
    - "x_3 x_4 + x_4 x_3 + a_{341} x_1^2 + a_{342} x_2^2 + a_{343} x_3^2 + a_{344} x_4^2"
  references:
    - "MR1356364"
    - "MR2580455"
  point_scheme: '20 points'
  centre: '\(\dim \operatorname{Z}(A)_2 = 4\) (the four squares are central)'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '-\mathrm{id}'
  nakayama_type: 'scalar'
  nakayama_constant: true
  homological_det: 1
  nakayama_field: 'gf-sample'
commutative:
  name: "commutative"
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 0
  hh0: [16, 60, 80]
  hh_qgr: [1, 15, 45, 35]
  point_scheme_dim: 3
  line_scheme_dim: 4
  line_scheme: 'all of \(\operatorname{Gr}(2,4)\) — every line of \(\mathbb{P}^3\) is a line module (the Klein quadric)'
  point_scheme: 'all of \(\mathbb{P}^3\)'
  year: "?"
  slug: "commutative"
  sortkey: "0000 commutative"
  ks_rank: 0
  ks_inj: true
  ks_surj: false
  centre_z2: 10
  centre_z3: 20
  centre_z4: 35
  normal_1: 3
  normal_2: 9
  introduced:
    reference: "MR917738"
    note: >
      The ordinary commutative polynomial ring \(k[x_1, x_2, x_3, x_4]\) — the
      basic Artin–Schelter regular algebra of dimension 4, included for comparison.
  construction: >
    The 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.
  relations:
    - "x_1 x_2 - x_2 x_1"
    - "x_1 x_3 - x_3 x_1"
    - "x_1 x_4 - x_4 x_1"
    - "x_2 x_3 - x_3 x_2"
    - "x_2 x_4 - x_4 x_2"
    - "x_3 x_4 - x_4 x_3"
  references:
    - "MR917738"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: true
  nakayama: '\mathrm{id}'
  nakayama_type: 'identity'
  homological_det: 1
  nakayama_field: 'qq'
deformed-skew-1:
  name: 'deformed skew \((x_3x_2;\ x_4^2)\)'
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 5
  hh0: [3, 4, 3]
  hh_qgr: [1, 2, 4, 7]
  year: "?"
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'two planes, a conic and two lines'
  point_scheme: 'three lines and a conic'
  slug: "deformed-skew-1"
  sortkey: "9999 deformed-skew-1"
  ks_rank: 4
  ks_inj: false
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "q_{13}, q_{14}, q_{23}, q_{24}"
      description: "skew-commutation parameters."
    - symbol: "t"
      description: "coefficient of the deforming term."
  introduced:
    reference: "2511.08390"
    note: >
      One of the deformed skew polynomial (\(\mathcal{F}\)) families; its earlier origin is not pinned down here.
  construction: >
    A quantum (skew) polynomial ring with a single quadratic term added: the
    \(x_3 x_2\) relation acquires \(-t\,x_4^2\).
  relations:
    - "x_2 x_1 - (q_{14}^2/q_{13}) x_1 x_2"
    - "x_3 x_1 - q_{13} x_1 x_3"
    - "x_4 x_1 - q_{14} x_1 x_4"
    - "x_3 x_2 - q_{23} x_2 x_3 - t x_4^2"
    - "x_4 x_2 - q_{24} x_2 x_4"
    - "x_4 x_3 - (1/q_{24}) x_3 x_4"
  references:

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(q_{14}^{-3},\tfrac{q_{14}^{2}}{q_{13}q_{23}q_{24}},q_{13}q_{23}q_{24},q_{14}\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
deformed-skew-2:
  name: 'deformed skew \((x_2x_1;\ x_3x_4)\)'
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 5
  hh0: [3, 4, 3]
  hh_qgr: [1, 2, 4, 7]
  year: "?"
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'two planes, two conics and two lines'
  point_scheme: 'a union of 5 lines'
  slug: "deformed-skew-2"
  sortkey: "9999 deformed-skew-2"
  ks_rank: 4
  ks_inj: false
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "q_{12}, q_{13}, q_{14}, q_{23}"
      description: "skew-commutation parameters."
    - symbol: "t"
      description: "coefficient of the deforming term."
  introduced:
    reference: "2511.08390"
    note: >
      One of the deformed skew polynomial (\(\mathcal{F}\)) families; its earlier origin is not pinned down here.
  construction: >
    A quantum (skew) polynomial ring with a single quadratic term added: the
    \(x_2 x_1\) relation acquires the cross-term \(-t\,x_3 x_4\).
  relations:
    - "x_2 x_1 - q_{12} x_1 x_2 - t x_3 x_4"
    - "x_3 x_1 - q_{13} x_1 x_3"
    - "x_4 x_1 - q_{14} x_1 x_4"
    - "x_3 x_2 - q_{23} x_2 x_3"
    - "x_4 x_2 - (1/(q_{13} q_{23} q_{14})) x_2 x_4"
    - "x_4 x_3 - (1/(q_{13} q_{23})) x_3 x_4"
  references:

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(\tfrac{1}{q_{12}q_{13}q_{14}},q_{12}q_{13}q_{14},q_{13}^{2}q_{23}^{2},\tfrac{1}{q_{13}^{2}q_{23}^{2}}\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
deformed-skew-3:
  name: 'deformed skew \((x_3x_2;\ x_4^2, x_1^2)\)'
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 5
  hh0: [2, 4, 8]
  hh_qgr: [1, 1, 3, 7]
  year: "?"
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'two planes, a quartic curve and two lines'
  point_scheme: 'a line, two conics and two points'
  slug: "deformed-skew-3"
  sortkey: "9999 deformed-skew-3"
  ks_rank: 3
  ks_inj: false
  ks_surj: false
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "q_{13}, q_{23}, q_{34}"
      description: "skew-commutation parameters."
    - symbol: "t, s"
      description: "coefficients of the two deforming terms."
  introduced:
    reference: "2511.08390"
    note: >
      One of the deformed skew polynomial (\(\mathcal{F}\)) families; its earlier origin is not pinned down here.
  construction: >
    A quantum (skew) polynomial ring with two quadratic terms added to one
    relation: \(x_3 x_2\) acquires \(-t\,x_4^2 - s\,x_1^2\).
  relations:
    - "x_2 x_1 - (1/q_{13}) x_1 x_2"
    - "x_3 x_1 - q_{13} x_1 x_3"
    - "x_4 x_1 - x_1 x_4"
    - "x_3 x_2 - q_{23} x_2 x_3 - t x_4^2 - s x_1^2"
    - "x_4 x_2 - (1/q_{34}) x_2 x_4"
    - "x_4 x_3 - q_{34} x_3 x_4"
  references:

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(1,\tfrac{q_{34}}{q_{13}q_{23}},\tfrac{q_{13}q_{23}}{q_{34}},1\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
deformed-skew-4:
  name: 'deformed skew \((x_3x_1, x_3x_2;\ x_4^2)\)'
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 4
  hh0: [2, 2, 2]
  hh_qgr: [1, 1, 8, 12]
  year: "?"
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a plane, two conics and three lines'
  point_scheme: 'a union of 3 lines'
  slug: "deformed-skew-4"
  sortkey: "9999 deformed-skew-4"
  ks_rank: 2
  ks_inj: false
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "q_{12}, q_{14}"
      description: "skew-commutation parameters."
    - symbol: "t, s"
      description: "coefficients of the two deforming terms."
  introduced:
    reference: "2511.08390"
    note: >
      One of the deformed skew polynomial (\(\mathcal{F}\)) families; its earlier origin is not pinned down here.
  construction: >
    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
    \(-s\,x_4^2\).
  relations:
    - "x_2 x_1 - q_{12} x_1 x_2"
    - "x_3 x_1 - (q_{14}^2/q_{12}) x_1 x_3 - t x_4^2"
    - "x_4 x_1 - q_{14} x_1 x_4"
    - "x_3 x_2 - (q_{14}^2 q_{12}) x_2 x_3 - s x_4^2"
    - "x_4 x_2 - q_{14} x_2 x_4"
    - "x_4 x_3 - (1/q_{14}) x_3 x_4"
  references:

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}(q_{14}^{-3},q_{14}^{-3},q_{14}^{5},q_{14})'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
deformed-skew-5:
  name: 'deformed skew \((x_2x_1, x_3x_2;\ x_3^2, x_4^2)\)'
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 4
  hh0: [2, 2, 2]
  hh_qgr: [1, 1, 8, 12]
  year: "?"
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a plane, a degree-5 curve, a conic and two lines'
  point_scheme: 'a line and a conic'
  slug: "deformed-skew-5"
  sortkey: "9999 deformed-skew-5"
  ks_rank: 2
  ks_inj: false
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "q_{12}, q_{24}"
      description: "skew-commutation parameters."
    - symbol: "t, s"
      description: "coefficients of the two deforming terms."
  introduced:
    reference: "2511.08390"
    note: >
      One of the deformed skew polynomial (\(\mathcal{F}\)) families; its earlier origin is not pinned down here.
  construction: >
    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 \(-s\,x_4^2\).
  relations:
    - "x_2 x_1 - q_{12} x_1 x_2 - t x_3^2"
    - "x_3 x_1 - (1/(q_{24}^6 q_{12})) x_1 x_3"
    - "x_4 x_1 - (1/q_{24}^3) x_1 x_4"
    - "x_3 x_2 - (q_{24}^6 q_{12}) x_2 x_3 - s x_4^2"
    - "x_4 x_2 - q_{24} x_2 x_4"
    - "x_4 x_3 - (1/q_{24}) x_3 x_4"
  references:

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}(q_{24}^{9},q_{24}^{-7},q_{24},q_{24}^{-3})'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
generalized-clifford-1:
  name: "Generalized Clifford 1"
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 1
  hh0: [1, 1, 7]
  hh_qgr: [1, 0, 16, 21]
  point_scheme_dim: 0
  line_scheme_dim: 1
  line_scheme: 'a quartic curve, four cubic curves and two conics'
  year: 2010
  slug: "generalized-clifford-1"
  sortkey: "2010 generalized-clifford-1"
  ks_rank: 1
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 2
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "α, β"
      description: 'Primitive 4th roots of unity (\(α^2 = β^2 = -1\)).'
    - symbol: "γ"
      description: "A nonzero scalar."
  introduced:
    reference: "MR2580455"
    note: >
      Example 1 of Cassidy–Vancliff. Of the three, it is a genuinely *skew* graded
      Clifford algebra: the generators \(q\)-commute by primitive 4th roots of
      unity. Its centre is trivial in degree 2 (central elements first appear in
      degree 4).
  construction: >
    A 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.
  relations:
    - "x_4 x_1 - α x_1 x_4"
    - "x_3 x_2 - β x_2 x_3"
    - "x_3^2 - x_1^2"
    - "x_4^2 - x_2^2"
    - "x_3 x_1 - x_1 x_3 + x_2^2"
    - "x_4 x_2 - x_2 x_4 + γ^2 x_1^2"
  references:
    - "MR2580455"
  point_scheme: '20 points'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: true
  nakayama: '\mathrm{id}'
  nakayama_type: 'identity'
  nakayama_constant: true
  homological_det: 1
  nakayama_field: 'gf-params'
generalized-clifford-2:
  name: "Generalized Clifford 2"
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 4
  hh0: [1, 9, 19]
  hh_qgr: [1, 0, 16, 21]
  point_scheme_dim: 0
  line_scheme_dim: 1
  line_scheme: 'two quartic curves and four cubic curves'
  year: 2010
  slug: "generalized-clifford-2"
  sortkey: "2010 generalized-clifford-2"
  ks_rank: 3
  ks_inj: false
  ks_surj: false
  centre_z2: 2
  centre_z3: 0
  centre_z4: 5
  normal_1: -1
  normal_2: 1
  parameters:
    - symbol: "α_1, α_2, β_1, β_2"
      description: "Four scalars."
  introduced:
    reference: "MR2580455"
    note: >
      Example 2 of Cassidy–Vancliff. It is the example that is an ordinary
      (non-skew) graded Clifford algebra and a complete intersection, and so has by
      far the largest centre of the three — a \(2\)-dimensional space of central
      quadrics, with centre Hilbert series \(1 + 2t^2 + 5t^4 + 8t^6 + \cdots\).
  construction: >
    Another 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.
  relations:
    - "x_3 x_1 + x_1 x_3 - β_2 x_2^2"
    - "x_4 x_1 + x_1 x_4 - α_2 x_3^2"
    - "x_2 x_3 - x_3 x_2"
    - "x_4^2 - x_2^2"
    - "x_4 x_2 + x_2 x_4 - x_3^2"
    - "α_1 x_3^2 + β_1 x_2^2 - x_1^2"
  references:
    - "MR2580455"
  point_scheme: '20 points'
  centre: '\(\dim \operatorname{Z}(A)_2 = 2\)'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '-\mathrm{id}'
  nakayama_type: 'scalar'
  nakayama_constant: true
  homological_det: 1
  nakayama_field: 'qq'
generalized-clifford-3:
  name: "Generalized Clifford 3"
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 4
  hh0: [1, 9, 7]
  point_scheme_dim: 0
  line_scheme_dim: 2
  line_scheme: 'an irreducible surface of degree 10 (at the 20-point regular parameter; read with care)'
  year: 2010
  slug: "generalized-clifford-3"
  sortkey: "2010 generalized-clifford-3"
  ks_rank: 4
  ks_inj: true
  ks_surj: false
  parameters:
    - symbol: "u_{13}, u_{14}, u_{24}, u_{34}"
      description: >
        Four scalars subject to a constraint variety for regularity
        (\(u_{34}^2 = 1\), \(u_{34} = u_{24}\), \(u_{14}^2 = u_{13}^2\)).
  introduced:
    reference: "MR2580455"
    note: >
      Example 3 of Cassidy–Vancliff. Like Example 1 it is a *skew* graded Clifford
      algebra, but it is Artin–Schelter regular only on a constraint subvariety of
      its parameter space; off that variety it fails to be regular.
  construction: >
    A 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.
  relations:
    - "x_1 x_3 + u_{13} x_3 x_1"
    - "x_1 x_4 + u_{14} x_4 x_1"
    - "x_3 x_4 + u_{34} x_4 x_3"
    - "x_4^2 - x_2^2"
    - "x_2 x_3 + x_3 x_2 + x_4^2"
    - "x_2 x_4 + u_{24} x_4 x_2 + x_1^2"
  references:
    - "MR2580455"
  point_scheme: '20 points'
  notes: >
    Artin–Schelter regular only at special (root-of-unity) values of the
    parameters; the invariants above are for such a point and depend on the choice
    (in particular the centre is not generic), so they should be read with care.

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}(1,-1,-1,-1)'
  nakayama_type: 'diagonal'
  homological_det: -1
  nakayama_field: 'qq'
goetz-kirkman-moore-vashaw-r:
  name: "Goetz–Kirkman–Moore–Vashaw R"
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 0
  hh0: [2, 12, 22]
  hh_qgr: [1, 3, 21, 23]
  line_scheme_dim: 2
  line_scheme: 'three quadric surfaces and four planes'
  year: 2024
  slug: "goetz-kirkman-moore-vashaw-r"
  sortkey: "2024 goetz-kirkman-moore-vashaw-r"
  ks_rank: 0
  ks_inj: true
  ks_surj: false
  centre_z2: 2
  centre_z3: 0
  centre_z4: 6
  parameters: []
  introduced:
    reference: "2410.08959"
    note: >
      The algebra "R" of Goetz–Kirkman–Moore–Vashaw, one of three dimension-four
      Artin–Schelter regular algebras they construct from a dual reflection group
      of order 16. It is a rigid (parameter-free) example.
  construction: >
    A 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.
  relations:
    - "x_1 x_2 + x_2 x_1"
    - "x_1 x_3 + x_4 x_2"
    - "x_1 x_4 - x_3 x_2"
    - "x_2 x_3 - x_4 x_1"
    - "x_2 x_4 + x_3 x_1"
    - "x_3 x_4 + x_4 x_3"
  references:
    - "2410.08959"

  point_scheme_dim: 1
  point_scheme: 'six lines'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '-\mathrm{id}'
  nakayama_type: 'scalar'
  homological_det: 1
  nakayama_field: 'qq'
goetz-kirkman-moore-vashaw-s:
  name: "Goetz–Kirkman–Moore–Vashaw S"
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 0
  hh0: [1, 4, 9]
  hh_qgr: [1, 0, 15, 20]
  line_scheme_dim: 1
  line_scheme: 'ten conics'
  year: 2024
  slug: "goetz-kirkman-moore-vashaw-s"
  sortkey: "2024 goetz-kirkman-moore-vashaw-s"
  ks_rank: 0
  ks_inj: true
  ks_surj: false
  centre_z2: 1
  centre_z3: 0
  centre_z4: 4
  parameters: []
  introduced:
    reference: "2410.08959"
    note: >
      The algebra "S" of Goetz–Kirkman–Moore–Vashaw, one of three dimension-four
      Artin–Schelter regular algebras they construct from a dual reflection group
      of order 16. It is a rigid (parameter-free) example.
  construction: >
    A 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.
  relations:
    - "x_1 x_2 - x_3 x_3"
    - "x_1 x_3 - x_2 x_4"
    - "x_1 x_4 - x_4 x_2"
    - "x_2 x_3 - x_3 x_1"
    - "x_3 x_2 - x_4 x_1"
    - "x_2 x_1 - x_4 x_4"
  references:
    - "2410.08959"

  point_scheme_dim: 0
  point_scheme: '20 points'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '-\mathrm{id}'
  nakayama_type: 'scalar'
  homological_det: 1
  nakayama_field: 'qq'
goetz-kirkman-moore-vashaw-t:
  name: "Goetz–Kirkman–Moore–Vashaw T"
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 0
  hh0: [1, 4, 9]
  hh_qgr: [1, 0, 15, 20]
  line_scheme_dim: 1
  line_scheme: 'ten conics'
  year: 2024
  slug: "goetz-kirkman-moore-vashaw-t"
  sortkey: "2024 goetz-kirkman-moore-vashaw-t"
  ks_rank: 0
  ks_inj: true
  ks_surj: false
  centre_z2: 0
  centre_z3: 0
  centre_z4: 2
  parameters: []
  introduced:
    reference: "2410.08959"
    note: >
      The algebra "T" of Goetz–Kirkman–Moore–Vashaw, one of three dimension-four
      Artin–Schelter regular algebras they construct from a dual reflection group
      of order 16. It is a rigid (parameter-free) example, differing from their
      algebra "S" in two signs but sharing its graded Hochschild cohomology.
  construction: >
    A 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.
  relations:
    - "x_1 x_2 - x_3 x_3"
    - "x_1 x_3 - x_2 x_4"
    - "x_1 x_4 + x_4 x_2"
    - "x_2 x_3 - x_3 x_1"
    - "x_3 x_2 - x_4 x_1"
    - "x_2 x_1 + x_4 x_4"
  references:
    - "2410.08959"

  point_scheme_dim: 0
  point_scheme: '20 points'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '-\mathrm{id}'
  nakayama_type: 'scalar'
  homological_det: 1
  nakayama_field: 'qq'
jordan:
  name: "Jordan"
  kind: "named"
  pym: true
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 1
  hh0: [2, 1, 1]
  hh_qgr: [1, 1, 0, 4]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a plane, two quartic curves, a cubic curve and a conic'
  year: 2016
  slug: "jordan"
  sortkey: "2016 jordan"
  ks_rank: 1
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  centre: 'trivial in low degrees (\(\dim \operatorname{Z}(A)_2 = \dim \operatorname{Z}(A)_3 = 0\)), as for a generic noncommutative \(\mathbb{P}^3\)'
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "a"
      description: "A single scalar parameter."
  introduced:
    reference: "MR3901632"
    note: >
      A member of the additive (\(\mathbb{G}_{\mathrm{a}}\)) family of regular algebras of
      Lecoutre–Sierra, obtained as a quantization of a Poisson structure.
  construction: >
    An "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.
  relations:
    - "(x_1 x_2 - x_2 x_1) + 4 a x_1^2"
    - "(x_2 x_3 - x_3 x_2) + 4 (a+1) x_2^2 - 8 (a+1)(a+2) x_1 x_2 - 4 (a+2) x_1 x_3"
    - "(x_1 x_3 - x_3 x_1) + 4 a x_1 x_2 + 8 a^2 x_1^2 - 8 a x_1^2"
    - "(x_2 x_4 - x_4 x_2) + 4 (a+1) x_2 x_3 + 8 a (a+1) x_2^2 - (64/3) a (a+1)(a+2) x_1 x_2 - 16 (a+1)(a+2) x_1 x_3 - 4 (a+3) x_1 x_4"
    - "(x_1 x_4 - x_4 x_1) + 4 a x_1 x_3 - 8 (a - a^2) x_1 x_2 - (64/6)(-a^3 + 3 a^2 - 2 a) x_1^2"
    - "(x_3 x_4 - x_4 x_3) + 4 (a+2) x_3^2 - 8 (a+2)(a+3) x_2 x_3 + (64/6)(a+2)(a+3)(a+4) x_1 x_3 - 4 (a+3) x_2 x_4 + 8 (a+3)(a+4) x_1 x_4"
  references:
    - "MR3901632"
  point_scheme: 'a line, a conic and a twisted cubic (a bouquet of rational normal curves)'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: 'x_1\mapsto x_1,\ x_2\mapsto(16a+20)x_1+x_2,\ x_3\mapsto(128a^{2}+288a+160)x_1+(16a+20)x_2+x_3,\ x_4\mapsto\tfrac{2048a^{3}+6144a^{2}+6016a+1920}{3}x_1+(128a^{2}+288a+160)x_2+(16a+20)x_3+x_4'
  nakayama_type: 'unipotent'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
ore-a:
  name: "A"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 1
  hh0: [2, 2, 2]
  hh_qgr: [1, 1, 8, 12]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a plane, a cubic curve and two lines'
  point_scheme: 'a union of 3 lines'
  year: 2009
  slug: "ore-a"
  sortkey: "2009 ore-a"
  ks_rank: 1
  ks_inj: true
  ks_surj: false
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family A among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_1 x_2 - x_2 x_1"
    - "x_4 x_3 - x_3 x_4 - x_3^2"
    - "x_3 x_1 - h x_1 x_3"
    - "x_3 x_2 - h (x_1 x_4 + x_2 x_3)"
    - "x_4 x_1 - h x_1 x_4"
    - "x_4 x_2 + h (2 x_2 x_3 + x_1 x_4 - x_2 x_4)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}(h^{-2},h^{-2},h^{2},h^{2})'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
ore-b:
  name: "B"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 1
  hh0: [2, 1, 0]
  hh_qgr: [1, 1, 17, 21]
  point_scheme_dim: 1
  line_scheme_dim: 1
  line_scheme: 'a curve of degree 20'
  point_scheme: 'a union of 2 lines'
  year: 2009
  slug: "ore-b"
  sortkey: "2009 ore-b"
  ks_rank: 1
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
    - symbol: "p"
      description: '\(p\) is a primitive 4th root of unity (\(p^2 = -1\))'
  introduced:
    reference: "MR2529094"
    note: "Family B among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 - p x_1 x_2"
    - "x_4 x_3 - p x_3 x_4"
    - "x_3 x_1 + h (- x_2 x_4)"
    - "x_3 x_2 + h (- x_1 x_4)"
    - "x_4 x_1 + h (x_2 x_3)"
    - "x_4 x_2 + h (- x_1 x_3)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}(-h^{-2},-h^{-2},-h^{2},-h^{2})'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'gf-params'
ore-c:
  name: "C"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 1
  hh0: [2, 1, 0]
  hh_qgr: [1, 1, 0, 4]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a quadric surface, a quartic curve, two conics and four lines'
  point_scheme: 'a regulus of 5 lines'
  year: 2009
  slug: "ore-c"
  sortkey: "2009 ore-c"
  ks_rank: 1
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: -1
  normal_2: -1
  parameters:
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
    - symbol: "p"
      description: '\(p\) is a primitive cube root of unity (\(p^2 + p + 1 = 0\))'
  introduced:
    reference: "MR2529094"
    note: "Family C among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_4 x_3 - p x_3 x_4"
    - "x_2 x_1 - p x_1 x_2"
    - "x_3 x_1 + h (x_1 x_3 - p^2 x_2 x_3 - x_1 x_4 + p x_2 x_4)"
    - "x_3 x_2 + h (p x_1 x_3 - x_2 x_3 - x_1 x_4 + p x_2 x_4)"
    - "x_4 x_1 + h (p x_1 x_3 + 2 p^2 x_2 x_3 - p x_1 x_4 + p x_2 x_4)"
    - "x_4 x_2 + h (p x_1 x_3 - p^2 x_2 x_3 - x_1 x_4 + x_2 x_4)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(-\tfrac{\omega}{3}h^{-2},-\tfrac{\omega}{3}h^{-2},-3\omega^{2}h^{2},-3\omega^{2}h^{2}\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'gf-params'
ore-d:
  name: "D"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 2
  hh0: [2, 2, 2]
  hh_qgr: [1, 1, 8, 12]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a plane and a curve of degree 11'
  point_scheme: 'a union of 4 lines'
  year: 2009
  slug: "ore-d"
  sortkey: "2009 ore-d"
  ks_rank: 2
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "p"
      description: "scalar parameter"
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family D among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_1 x_2 + x_2 x_1"
    - "x_4 x_3 - p x_3 x_4"
    - "x_3 x_1 + h (p x_1 x_3)"
    - "x_3 x_2 + h (p^2 x_2 x_3 - x_1 x_4)"
    - "x_4 x_1 - h (p x_1 x_4)"
    - "x_4 x_2 + h (- x_1 x_3 - x_2 x_4)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}(p^{-2}h^{-2},p^{-2}h^{-2},p^{2}h^{2},p^{2}h^{2})'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
ore-e:
  name: "E"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 1
  hh0: [2, 1, 0]
  hh_qgr: [1, 2, 3, 6]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'two quadric surfaces, two planes and two lines'
  point_scheme: 'a union of 6 lines'
  year: 2009
  slug: "ore-e"
  sortkey: "2009 ore-e"
  ks_rank: 1
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
    - symbol: "p"
      description: '\(p\) is a primitive 4th root of unity (\(p^2 = -1\))'
  introduced:
    reference: "MR2529094"
    note: "Family E among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 + x_1 x_2"
    - "x_4 x_3 - p x_3 x_4"
    - "x_3 x_1 + h (- x_1 x_4 - x_2 x_4)"
    - "x_3 x_2 + h (- x_1 x_4 + x_2 x_4)"
    - "x_4 x_1 + h (x_1 x_3 - x_2 x_3)"
    - "x_4 x_2 + h (- x_1 x_3 - x_2 x_3)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: 'x_1\mapsto-\tfrac{i}{2}h^{-2}x_2,\ x_2\mapsto\tfrac{i}{2}h^{-2}x_1,\ x_3\mapsto-2i\,h^{2}x_3,\ x_4\mapsto2i\,h^{2}x_4'
  nakayama_type: 'monomial'
  nakayama_constant: false
  homological_det: -1
  nakayama_field: 'gf-params'
ore-extension-of-commutative:
  name: "Ore extension of commutative"
  kind: "named"
  pym: true
  poisson: true
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 6
  hh0: [4, 9, 9]
  hh_qgr: [1, 3, 6, 8]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a double plane (a plane with multiplicity two)'
  year: 2015
  slug: "ore-extension-of-commutative"
  sortkey: "2015 ore-extension-of-commutative"
  ks_rank: 9
  ks_inj: false
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: -1
  normal_2: -1
  parameters:
    - symbol: "c_1, c_2, c_3, d_1, d_2, d_3"
      description: 'Six scalars; \(b_i = -c_{i-1} - 2\) are determined by the \(c_i\).'
  introduced:
    reference: "MR3366864"
    note: >
      One of Pym's quantum deformations of projective 3-space, arising from a
      quadratic Poisson structure (the type \(S(2,3)\)).
  construction: >
    A 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.
  relations:
    - "x_4 x_1 - x_1 x_4 - x_1^2 - x_1 ((-c_3 - 2) x_2 + c_1 x_3) - d_1 x_2 x_3"
    - "x_4 x_2 - x_2 x_4 - x_2^2 - x_2 ((-c_1 - 2) x_3 + c_2 x_1) - d_2 x_3 x_1"
    - "x_4 x_3 - x_3 x_4 - x_3^2 - x_3 ((-c_2 - 2) x_1 + c_3 x_2) - d_3 x_1 x_2"
    - "x_2 x_3 - x_3 x_2"
    - "x_3 x_1 - x_1 x_3"
    - "x_1 x_2 - x_2 x_1"
  references:
    - "MR3366864"
  point_scheme: 'three lines and a rational normal quartic'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: true
  nakayama: '\mathrm{id}'
  nakayama_type: 'identity'
  nakayama_constant: true
  homological_det: 1
  nakayama_field: 'gf-sample'
ore-f:
  name: "F"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 1
  hh0: [2, 1, 0]
  hh_qgr: [1, 1, 16, 20]
  point_scheme_dim: 1
  line_scheme_dim: 1
  line_scheme: 'a curve of degree 20'
  point_scheme: 'a union of 2 lines'
  year: 2009
  slug: "ore-f"
  sortkey: "2009 ore-f"
  ks_rank: 1
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
    - symbol: "p"
      description: '\(p\) is a primitive 4th root of unity (\(p^2 = -1\))'
  introduced:
    reference: "MR2529094"
    note: "Family F among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 + x_1 x_2"
    - "x_4 x_3 - p x_3 x_4"
    - "x_3 x_1 + h (x_1 x_3 + p x_2 x_3 - x_1 x_4 + x_2 x_4)"
    - "x_3 x_2 + h (p x_1 x_3 - x_2 x_3 - x_1 x_4 - x_2 x_4)"
    - "x_4 x_1 + h (p x_1 x_3 - p x_2 x_3 - p x_1 x_4 - x_2 x_4)"
    - "x_4 x_2 + h (p x_1 x_3 + p x_2 x_3 - x_1 x_4 + p x_2 x_4)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(-\tfrac{i}{2}h^{-2},-\tfrac{i}{2}h^{-2},2i\,h^{2},2i\,h^{2}\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'gf-params'
ore-g:
  name: "G"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 3
  hh0: [2, 2, 2]
  hh_qgr: [1, 1, 8, 12]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a plane and a curve of degree 11'
  point_scheme: 'a union of 4 lines'
  year: 2009
  slug: "ore-g"
  sortkey: "2009 ore-g"
  ks_rank: 2
  ks_inj: false
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "p"
      description: "scalar parameter"
    - symbol: "f"
      description: "scalar parameter"
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family G among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 - x_1 x_2"
    - "x_4 x_3 - p x_3 x_4"
    - "x_3 x_1 + h (- p x_1 x_3)"
    - "x_3 x_2 + h (- p x_1 x_3 - p^2 x_2 x_3 - x_1 x_4)"
    - "x_4 x_1 + h (- p x_1 x_4)"
    - "x_4 x_2 + h (- f x_1 x_3 + x_1 x_4 - x_2 x_4)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}(p^{-2}h^{-2},p^{-2}h^{-2},p^{2}h^{2},p^{2}h^{2})'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
ore-h:
  name: "H"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 2
  hh0: [3, 3, 1]
  hh_qgr: [1, 3, 6, 8]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a quadric surface and three planes, one with multiplicity two (non-reduced; scheme degree 6)'
  point_scheme: 'a union of 4 lines'
  year: 2009
  slug: "ore-h"
  sortkey: "2009 ore-h"
  ks_rank: 2
  ks_inj: true
  ks_surj: false
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "f"
      description: "scalar parameter"
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family H among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 - x_1 x_2 - x_1^2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 - h x_1 x_4"
    - "x_3 x_2 - h f x_1 x_4 - h x_2 x_4"
    - "x_4 x_1 - h x_1 x_3"
    - "x_4 x_2 - h f x_1 x_3 - h x_2 x_3"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: 'x_1\mapsto h^{-2}x_1,\ x_2\mapsto h^{-2}\bigl((2-2f)x_1+x_2\bigr),\ x_3\mapsto-h^{2}x_3,\ x_4\mapsto-h^{2}x_4'
  nakayama_type: 'general'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
ore-i:
  name: "I"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 1
  hh0: [2, 1, 0]
  hh_qgr: [1, 1, 16, 20]
  point_scheme_dim: 1
  line_scheme_dim: 1
  line_scheme: 'a curve of degree 20'
  point_scheme: 'a union of 2 lines'
  year: 2009
  slug: "ore-i"
  sortkey: "2009 ore-i"
  ks_rank: 1
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
    - symbol: "q"
      description: '\(q\) is a primitive 4th root of unity (\(q^2 = -1\))'
  introduced:
    reference: "MR2529094"
    note: "Family I among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 - q x_1 x_2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 + h (q x_1 x_3 + q x_2 x_3 - x_1 x_4 + q x_2 x_4)"
    - "x_3 x_2 + h (- x_1 x_3 - x_2 x_3 - x_1 x_4 + q x_2 x_4)"
    - "x_4 x_1 + h (- x_1 x_3 - q x_2 x_3 - q x_1 x_4 + q x_2 x_4)"
    - "x_4 x_2 + h (x_1 x_3 + q x_2 x_3 - x_1 x_4 + x_2 x_4)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(-\tfrac{i}{2}h^{-2},-\tfrac{i}{2}h^{-2},2i\,h^{2},2i\,h^{2}\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'gf-params'
ore-j:
  name: "J"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 1
  hh0: [2, 1, 0]
  hh_qgr: [1, 2, 3, 6]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'two quadric surfaces, two planes and two lines'
  point_scheme: 'a union of 6 lines'
  year: 2009
  slug: "ore-j"
  sortkey: "2009 ore-j"
  ks_rank: 1
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
    - symbol: "q"
      description: '\(q\) is a primitive 4th root of unity (\(q^2 = -1\))'
  introduced:
    reference: "MR2529094"
    note: "Family J among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 - q x_1 x_2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 + h (- x_2 x_3 - x_2 x_4)"
    - "x_3 x_2 + h (x_1 x_3 - x_1 x_4)"
    - "x_4 x_1 + h (- x_2 x_3 + x_2 x_4)"
    - "x_4 x_2 + h (- x_1 x_3 - x_1 x_4)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: 'x_1\mapsto-\tfrac{i}{2}h^{-2}x_1,\ x_2\mapsto\tfrac{i}{2}h^{-2}x_2,\ x_3\mapsto2i\,h^{2}x_4,\ x_4\mapsto-2i\,h^{2}x_3'
  nakayama_type: 'monomial'
  nakayama_constant: false
  homological_det: -1
  nakayama_field: 'gf-params'
ore-k:
  name: "K"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 3
  hh0: [3, 3, 1]
  hh_qgr: [1, 2, 1, 4]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a surface of degree 2 and a curve of degree 6'
  point_scheme: 'a union of 6 lines'
  year: 2009
  slug: "ore-k"
  sortkey: "2009 ore-k"
  ks_rank: 3
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "f"
      description: "scalar parameter"
    - symbol: "q"
      description: "scalar parameter"
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family K among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 - q x_1 x_2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 + h (- x_1 x_3)"
    - "x_3 x_2 + h (- x_2 x_4)"
    - "x_4 x_1 + h (- x_1 x_4)"
    - "x_4 x_2 + h (- f x_2 x_3)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: 'x_1\mapsto\tfrac{1}{qh^{2}}x_1,\ x_2\mapsto\tfrac{q}{fh^{2}}x_2,\ x_3\mapsto-h^{2}x_4,\ x_4\mapsto-fh^{2}x_3'
  nakayama_type: 'monomial'
  nakayama_constant: false
  homological_det: -1
  nakayama_field: 'qq'
ore-l:
  name: "L"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 3
  hh0: [3, 3, 1]
  hh_qgr: [1, 3, 6, 8]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a quadric surface and four planes'
  point_scheme: 'a union of 6 lines'
  year: 2009
  slug: "ore-l"
  sortkey: "2009 ore-l"
  ks_rank: 3
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "f"
      description: "scalar parameter"
    - symbol: "q"
      description: "scalar parameter"
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family L among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 - q x_1 x_2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 + h (- f x_1 x_4)"
    - "x_3 x_2 + h (- x_2 x_4)"
    - "x_4 x_1 + h (- f x_1 x_3)"
    - "x_4 x_2 + h (- x_2 x_3)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(\tfrac{1}{f^{2}qh^{2}},\tfrac{q}{h^{2}},-fh^{2},-fh^{2}\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
ore-m:
  name: "M"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 2
  hh0: [2, 2, 2]
  hh_qgr: [1, 3, 9, 11]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a surface of degree 4'
  point_scheme: 'a union of 6 lines'
  year: 2009
  slug: "ore-m"
  sortkey: "2009 ore-m"
  ks_rank: 2
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "f"
      description: "scalar parameter"
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family M among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 + x_1 x_2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 + h (- x_2 x_3 - x_1 x_4)"
    - "x_3 x_2 + h (- f x_1 x_3 + x_2 x_4)"
    - "x_4 x_1 + h (- x_1 x_3 + x_2 x_4)"
    - "x_4 x_2 + h (x_2 x_3 + f x_1 x_4)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(\tfrac{1}{(f-1)h^{2}},\tfrac{1}{(f-1)h^{2}},(1-f)h^{2},(1-f)h^{2}\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
ore-n:
  name: "N"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 3
  hh0: [2, 2, 2]
  hh_qgr: [1, 1, 15, 19]
  point_scheme_dim: 1
  line_scheme_dim: 1
  line_scheme: 'a curve of degree 20'
  point_scheme: 'a union of 2 lines'
  year: 2009
  slug: "ore-n"
  sortkey: "2009 ore-n"
  ks_rank: 2
  ks_inj: false
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "f"
      description: "scalar parameter"
    - symbol: "g"
      description: "scalar parameter"
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family N among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 + x_1 x_2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 + h (g x_2 x_3 - f x_2 x_4)"
    - "x_3 x_2 + h (- g x_1 x_3 - f x_1 x_4)"
    - "x_4 x_1 + h (- f x_2 x_3 + g x_2 x_4)"
    - "x_4 x_2 + h (- f x_1 x_3 - g x_1 x_4)"
  notes: 'Constraint: \(f^2 \neq g^2\).'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(\tfrac{1}{(g^{2}-f^{2})h^{2}},\tfrac{1}{(g^{2}-f^{2})h^{2}},(g^{2}-f^{2})h^{2},(g^{2}-f^{2})h^{2}\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
ore-o:
  name: "O"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 2
  hh0: [2, 2, 2]
  hh_qgr: [1, 3, 9, 11]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a surface of degree 8'
  point_scheme: 'a union of 6 lines'
  year: 2009
  slug: "ore-o"
  sortkey: "2009 ore-o"
  ks_rank: 2
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "f"
      description: "scalar parameter"
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family O among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 + x_1 x_2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 + h (- x_1 x_3 - f x_2 x_4)"
    - "x_3 x_2 + h (x_2 x_3 - x_1 x_4)"
    - "x_4 x_1 + h (- f x_2 x_3 + x_1 x_4)"
    - "x_4 x_2 + h (- x_1 x_3 - x_2 x_4)"
  notes: 'Constraint: \(f \neq 1\).'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(\tfrac{1}{(1-f)h^{2}},\tfrac{1}{(1-f)h^{2}},(1-f)h^{2},(1-f)h^{2}\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
ore-p:
  name: "P"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 2
  hh0: [2, 2, 2]
  hh_qgr: [1, 1, 15, 19]
  point_scheme_dim: 1
  line_scheme_dim: 1
  line_scheme: 'a curve of degree 20'
  point_scheme: 'a union of 2 lines'
  year: 2009
  slug: "ore-p"
  sortkey: "2009 ore-p"
  ks_rank: 2
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "f"
      description: "scalar parameter"
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family P among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 + x_1 x_2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 + h (- x_1 x_4 - f x_2 x_4)"
    - "x_3 x_2 + h (- x_1 x_4 - x_2 x_4)"
    - "x_4 x_1 + h (- x_1 x_3 + f x_2 x_3)"
    - "x_4 x_2 + h (x_1 x_3 - x_2 x_3)"
  notes: 'Constraint: \(f \neq 1\).'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(\tfrac{1}{(f-1)h^{2}},\tfrac{1}{(f-1)h^{2}},(f-1)h^{2},(f-1)h^{2}\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
ore-q:
  name: "Q"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 1
  hh0: [2, 1, 0]
  hh_qgr: [1, 1, 1, 5]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'two quadric surfaces, a plane and a cubic curve'
  point_scheme: 'a union of 6 lines'
  year: 2009
  slug: "ore-q"
  sortkey: "2009 ore-q"
  ks_rank: 1
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family Q among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 + x_1 x_2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 + h (- x_1 x_4)"
    - "x_3 x_2 + h (- x_1 x_3 - x_2 x_3 - x_1 x_4)"
    - "x_4 x_1 + h (x_1 x_3)"
    - "x_4 x_2 + h (- x_1 x_3 + x_1 x_4 - x_2 x_4)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: 'x_1\mapsto h^{-2}x_1,\ x_2\mapsto-h^{-2}x_2,\ x_3\mapsto-h^{2}x_4,\ x_4\mapsto h^{2}x_3'
  nakayama_type: 'monomial'
  nakayama_constant: false
  homological_det: -1
  nakayama_field: 'qq'
ore-r:
  name: "R"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 1
  hh0: [2, 1, 0]
  hh_qgr: [1, 2, 6, 9]
  point_scheme_dim: 2
  line_scheme_dim: 3
  line_scheme: 'a quadric threefold and two conics'
  point_scheme: 'a quadric surface'
  year: 2009
  slug: "ore-r"
  sortkey: "2009 ore-r"
  ks_rank: 1
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family R among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 + x_1 x_2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 + h (- x_1 x_3 - x_2 x_3 - x_1 x_4)"
    - "x_3 x_2 + h (- x_1 x_4)"
    - "x_4 x_1 + h (- x_2 x_3)"
    - "x_4 x_2 + h (x_2 x_3 + x_1 x_4 - x_2 x_4)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: 'x_1\mapsto h^{-2}x_2,\ x_2\mapsto-h^{-2}x_1,\ x_3\mapsto-h^{2}x_4,\ x_4\mapsto h^{2}x_3'
  nakayama_type: 'monomial'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
ore-s:
  name: "S"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 1
  hh0: [2, 1, 0]
  hh_qgr: [1, 1, 17, 21]
  point_scheme_dim: 1
  line_scheme_dim: 1
  line_scheme: 'four conics and twelve lines'
  point_scheme: 'a union of 2 lines'
  year: 2009
  slug: "ore-s"
  sortkey: "2009 ore-s"
  ks_rank: 1
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family S among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 + x_1 x_2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 + h (x_1 x_3 - x_2 x_3 - x_1 x_4 - x_2 x_4)"
    - "x_3 x_2 + h (- x_1 x_3 + x_2 x_3 - x_1 x_4 - x_2 x_4)"
    - "x_4 x_1 + h (- x_1 x_3 - x_2 x_3 + x_1 x_4 - x_2 x_4)"
    - "x_4 x_2 + h (- x_1 x_3 - x_2 x_3 - x_1 x_4 + x_2 x_4)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(-\tfrac{1}{4}h^{-2},-\tfrac{1}{4}h^{-2},-4h^{2},-4h^{2}\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
ore-t:
  name: "T"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 1
  hh0: [2, 1, 0]
  hh_qgr: [1, 2, 8, 11]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a surface of degree 3 and a curve of degree 5'
  point_scheme: 'two lines and two conics'
  year: 2009
  slug: "ore-t"
  sortkey: "2009 ore-t"
  ks_rank: 1
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family T among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 + x_1 x_2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 + h (x_1 x_3 - x_2 x_3 - x_1 x_4 - x_2 x_4)"
    - "x_3 x_2 + h (- x_1 x_3 + x_2 x_3 - x_1 x_4 - x_2 x_4)"
    - "x_4 x_1 + h (- x_1 x_3 - x_2 x_3 - x_1 x_4 + x_2 x_4)"
    - "x_4 x_2 + h (- x_1 x_3 - x_2 x_3 + x_1 x_4 - x_2 x_4)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: 'x_1\mapsto-\tfrac{1}{4}h^{-2}x_2,\ x_2\mapsto-\tfrac{1}{4}h^{-2}x_1,\ x_3\mapsto-4h^{2}x_3,\ x_4\mapsto-4h^{2}x_4'
  nakayama_type: 'monomial'
  nakayama_constant: false
  homological_det: -1
  nakayama_field: 'qq'
ore-u:
  name: "U"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 1
  hh0: [2, 1, 0]
  hh_qgr: [1, 2, 8, 11]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a surface of degree 3 and a curve of degree 5'
  point_scheme: 'two lines and two conics'
  year: 2009
  slug: "ore-u"
  sortkey: "2009 ore-u"
  ks_rank: 1
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family U among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 + x_1 x_2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 + h (x_1 x_3 - x_2 x_3 - x_1 x_4 - x_2 x_4)"
    - "x_3 x_2 + h (- x_1 x_3 - x_2 x_3 - x_1 x_4 + x_2 x_4)"
    - "x_4 x_1 + h (- x_1 x_3 - x_2 x_3 + x_1 x_4 - x_2 x_4)"
    - "x_4 x_2 + h (- x_1 x_3 + x_2 x_3 - x_1 x_4 - x_2 x_4)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: 'x_1\mapsto-\tfrac{1}{4}h^{-2}x_1,\ x_2\mapsto-\tfrac{1}{4}h^{-2}x_2,\ x_3\mapsto-4h^{2}x_4,\ x_4\mapsto-4h^{2}x_3'
  nakayama_type: 'monomial'
  nakayama_constant: false
  homological_det: -1
  nakayama_field: 'qq'
ore-v:
  name: "V"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 1
  hh0: [2, 1, 0]
  hh_qgr: [1, 1, 1, 5]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'two quadric surfaces, a plane and a cubic curve'
  point_scheme: 'a union of 6 lines'
  year: 2009
  slug: "ore-v"
  sortkey: "2009 ore-v"
  ks_rank: 1
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family V among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 - x_1 x_2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 + h (- x_2 x_3 - x_1 x_4)"
    - "x_3 x_2 - h x_2 x_3"
    - "x_4 x_1 + h (x_1 x_3 - x_2 x_3)"
    - "x_4 x_2 - h x_2 x_4"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: 'x_1\mapsto h^{-2}(-x_1+x_2),\ x_2\mapsto h^{-2}x_2,\ x_3\mapsto-h^{2}x_4,\ x_4\mapsto h^{2}x_3'
  nakayama_type: 'general'
  nakayama_constant: false
  homological_det: -1
  nakayama_field: 'qq'
ore-w:
  name: "W"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 2
  hh0: [2, 2, 2]
  hh_qgr: [1, 1, 15, 19]
  point_scheme_dim: 1
  line_scheme_dim: 1
  line_scheme: 'a curve of degree 20'
  point_scheme: 'a union of 2 lines'
  year: 2009
  slug: "ore-w"
  sortkey: "2009 ore-w"
  ks_rank: 2
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "f"
      description: "scalar parameter; constraint f != -1"
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family W among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 - x_1 x_2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 + h (- f x_2 x_3 - x_1 x_4)"
    - "x_3 x_2 + h (- x_1 x_3 + x_2 x_4)"
    - "x_4 x_1 + h (- x_1 x_3 - f x_2 x_4)"
    - "x_4 x_2 + h (x_2 x_3 - x_1 x_4)"
  notes: "Constraint: f != -1."

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(\tfrac{1}{(f+1)h^{2}},\tfrac{1}{(f+1)h^{2}},(f+1)h^{2},(f+1)h^{2}\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
ore-x:
  name: "X"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 1
  hh0: [3, 3, 1]
  hh_qgr: [1, 3, 6, 8]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a quadric surface and three planes, one with multiplicity two (non-reduced; scheme degree 6)'
  point_scheme: 'a union of 4 lines'
  year: 2009
  slug: "ore-x"
  sortkey: "2009 ore-x"
  ks_rank: 1
  ks_inj: true
  ks_surj: false
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family X among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 - x_1 x_2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 + h (- x_1 x_4)"
    - "x_3 x_2 + h (- x_1 x_4 - x_2 x_4)"
    - "x_4 x_1 + h (- x_1 x_3)"
    - "x_4 x_2 + h (- x_1 x_3 - x_2 x_3)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: 'x_1\mapsto h^{-2}x_1,\ x_2\mapsto h^{-2}(-2x_1+x_2),\ x_3\mapsto-h^{2}x_3,\ x_4\mapsto-h^{2}x_4'
  nakayama_type: 'general'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
ore-y:
  name: "Y"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 2
  hh0: [2, 3, 4]
  hh_qgr: [1, 1, 16, 20]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a plane and six lines'
  point_scheme: 'a union of 4 lines'
  year: 2009
  slug: "ore-y"
  sortkey: "2009 ore-y"
  ks_rank: 1
  ks_inj: false
  ks_surj: false
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "f"
      description: "scalar parameter"
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family Y among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 - x_1 x_2"
    - "x_4 x_3 + x_3 x_4"
    - "x_3 x_1 + h (- x_1 x_3)"
    - "x_3 x_2 + h (- f x_1 x_3 + x_2 x_3 - x_1 x_4)"
    - "x_4 x_1 + h (- x_1 x_4)"
    - "x_4 x_2 + h (- x_1 x_3 - f x_1 x_4 + x_2 x_4)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}(h^{-2},h^{-2},h^{2},h^{2})'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
ore-z:
  name: "Z"
  kind: "double-Ore"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 2
  hh0: [2, 2, 2]
  hh_qgr: [1, 1, 15, 19]
  point_scheme_dim: 1
  line_scheme_dim: 1
  line_scheme: 'a curve of degree 20'
  point_scheme: 'a union of 2 lines'
  year: 2009
  slug: "ore-z"
  sortkey: "2009 ore-z"
  ks_rank: 2
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "f"
      description: "scalar parameter; constraint f(1+f) != 0"
    - symbol: "h"
      description: "deformation parameter (generic scalar)"
  introduced:
    reference: "MR2529094"
    note: "Family Z among the double extension regular algebras of type (14641) classified by Zhang–Zhang."
  relations:
    - "x_2 x_1 + x_1 x_2"
    - "x_4 x_3 - x_3 x_4"
    - "x_3 x_1 + h (- x_1 x_3 - x_2 x_4)"
    - "x_3 x_2 + h (- x_2 x_3 - x_1 x_4)"
    - "x_4 x_1 + h (- f x_2 x_3 + x_1 x_4)"
    - "x_4 x_2 + h (- f x_1 x_3 + x_2 x_4)"
  notes: 'Constraint: \(f(1+f) \neq 0\).'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(\tfrac{1}{(f+1)h^{2}},\tfrac{1}{(f+1)h^{2}},(f+1)h^{2},(f+1)h^{2}\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
s-d-i:
  name: '\(\mathrm{S}_{d,i}\)'
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 3
  hh0: [1, 3, 9]
  hh_qgr: [1, 0, 4, 9]
  line_scheme_dim: 1
  line_scheme: 'a curve of degree 20'
  year: 2016
  slug: "s-d-i"
  sortkey: "2016 s-d-i"
  ks_rank: 3
  ks_inj: true
  ks_surj: true
  centre_z2: 1
  centre_z3: 0
  centre_z4: 2
  parameters:
    - symbol: "β, γ"
      description: >
        With \(α = -(β + γ)/(1 + βγ)\); \(α, β, γ\) nonzero and not \(\pm 1\).
    - symbol: "d = (d_1, d_2)"
      description: >
        A point of \(\mathbb{P}^1\); these are the coefficients of the two central
        quadrics \(\Omega_1, \Omega_2\) in the replaced relation.
  introduced:
    reference: "MR3490085"
    note: >
      One of Davies' algebras \(\mathrm{S}_{d,i}\) (equation 6.1.4), a finite member
      of the one-parameter family of twisted Sklyanin algebras of which
      \(\mathrm{S}_\infty\) is the \(d \to \infty\) limit.
  construction: >
    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
    \(\mathrm{S}_\infty\).
  relations:
    - "(x_1 x_2 - x_2 x_1) - α (x_3 x_4 + x_4 x_3)"
    - "(x_1 x_2 + x_2 x_1) - (x_3 x_4 - x_4 x_3)"
    - "(x_1 x_3 - x_3 x_1) - β (x_4 x_2 + x_2 x_4)"
    - "(x_1 x_3 + x_3 x_1) - (x_4 x_2 - x_2 x_4)"
    - "(x_1 x_4 - x_4 x_1) - γ (x_2 x_3 + x_3 x_2)"
    - "(x_1 x_4 + x_4 x_1) - (x_2 x_3 - x_3 x_2)"
    - "d_1 (-x_1^2 + x_2^2 + x_3^2 + x_4^2) + d_2 (x_2^2 + ((1+α)/(1-β)) x_3^2 + ((1-α)/(1+γ)) x_4^2)"
  references:
    - "MR3490085"
    - "MR3885145"
  notes: >
    This page represents the general construction. For a chosen index
    \(i \in \{1, \dots, 6\}\), the algebra \(\mathrm{S}_{d,i}\) has the six relations
    \(f_1, \dots, f_6\) of the Sklyanin algebra (the first six relations above) with
    \(f_i\) deleted and the quadric \(d_1 \Omega_1 + d_2 \Omega_2\) (the last
    relation above) imposed in its place — so exactly six relations in total.
  code_derive:
    - "alpha = -(beta + gamma)/(1 + beta*gamma)"

  point_scheme_dim: 1
  point_scheme: 'a quartic curve and 4 points'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}(-1,-1,1,1)'
  nakayama_type: 'diagonal'
  nakayama_constant: true
  homological_det: 1
  nakayama_field: 'qq'
s-d-i-twist:
  name: '\(\mathrm{S}_{d,i}\) twist'
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 3
  hh0: [1, 8, 17]
  hh_qgr: [1, 0, 14, 19]
  line_scheme_dim: 1
  line_scheme: 'a degree-12 curve, a quartic curve and two conics'
  year: 2016
  slug: "s-d-i-twist"
  sortkey: "2016 s-d-i-twist"
  ks_rank: 3
  ks_inj: true
  ks_surj: false
  parameters:
    - symbol: "β, γ"
      description: >
        With \(α = -(β + γ)/(1 + βγ)\); \(α, β, γ\) nonzero and not \(\pm 1\).
    - symbol: "d = (d_1, d_2)"
      description: >
        A point of \(\mathbb{P}^1\); the coefficients of the two twisted quadrics
        \(\Omega'_1, \Omega'_2\) in the replaced relation.
  introduced:
    reference: "MR3490085"
    note: >
      The cocycle twist of Davies' \(\mathrm{S}_{d,i}\) (equation 6.1.8 / 6.1.4
      twisted) by the Klein four-group; a finite member of the family of which the
      \(\mathrm{S}_\infty\) twist is the \(d \to \infty\) limit.
  construction: >
    As 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.
  relations:
    - "(x_1 x_2 - x_2 x_1) - α (x_3 x_4 - x_4 x_3)"
    - "(x_1 x_2 + x_2 x_1) - (x_3 x_4 + x_4 x_3)"
    - "(x_1 x_3 - x_3 x_1) - β (x_4 x_2 - x_2 x_4)"
    - "(x_1 x_3 + x_3 x_1) - (x_4 x_2 + x_2 x_4)"
    - "(x_1 x_4 - x_4 x_1) + γ (x_2 x_3 - x_3 x_2)"
    - "(x_1 x_4 + x_4 x_1) + (x_2 x_3 + x_3 x_2)"
    - "d_1 (-x_1^2 + x_2^2 + x_3^2 - x_4^2) + d_2 (x_2^2 + ((1+α)/(1-β)) x_3^2 - ((1-α)/(1+γ)) x_4^2)"
  references:
    - "MR3490085"
    - "MR3885145"
  notes: >
    This page represents the general construction. For a chosen index
    \(i \in \{1, \dots, 6\}\), the algebra has the six twisted Sklyanin relations
    \(f'_1, \dots, f'_6\) (the first six above) with \(f'_i\) deleted and the
    twisted quadric \(d_1 \Omega'_1 + d_2 \Omega'_2\) (the last relation above)
    imposed in its place — so exactly six relations in total. For AS-regularity one
    excludes the degenerate \((d_1, d_2)\) noted in the source (e.g. for \(i = 1\),
    \((d_1, d_2) \neq (1, 0), (1, -1 - βγ)\)).
  code_derive:
    - "alpha = -(beta + gamma)/(1 + beta*gamma)"

  point_scheme_dim: 0
  point_scheme: '20 points'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '-\mathrm{id}'
  nakayama_type: 'scalar'
  nakayama_constant: true
  homological_det: 1
  nakayama_field: 'qq'
s-infinity:
  name: '\(\mathrm{S}_\infty\)'
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 2
  hh0: [1, 2, 5]
  hh_qgr: [1, 0, 2, 7]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'the surface of secant lines to the point-scheme quartic elliptic curve \(E\) (the same curve as Sklyanin) — an elliptic ruled surface of degree 8'
  year: 2016
  slug: "s-infinity"
  sortkey: "2016 s-infinity"
  ks_rank: 2
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 2
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "β, γ"
      description: >
        With \(α = -(β + γ)/(1 + βγ)\); \(α, β, γ\) nonzero and not \(\pm 1\).
  introduced:
    reference: "MR3490085"
    note: >
      One of the algebras \(\mathrm{S}_\infty\) arising in Davies' study of cocycle
      twists of the 4-dimensional Sklyanin algebra; an exotic elliptic algebra in
      the sense of Chirvasitu–Smith. The subscript \(\infty\) marks it as the
      degenerate limit of Davies' one-parameter family \(\mathrm{S}_d\) of twisted
      Sklyanin algebras (the \(d \to \infty\) member), where the two central
      quadrics of the Sklyanin algebra are instead imposed as defining relations.
  construction: >
    \(\mathrm{S}_\infty\) 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.
  relations:
    - "(x_1 x_2 - x_2 x_1) - α (x_3 x_4 + x_4 x_3)"
    - "(x_1 x_2 + x_2 x_1) - (x_3 x_4 - x_4 x_3)"
    - "(x_1 x_3 - x_3 x_1) - β (x_4 x_2 + x_2 x_4)"
    - "(x_1 x_3 + x_3 x_1) - (x_4 x_2 - x_2 x_4)"
    - "-x_1^2 + x_2^2 + x_3^2 + x_4^2"
    - "x_2^2 + ((1+α)/(1-β)) x_3^2 + ((1-α)/(1+γ)) x_4^2"
  references:
    - "MR3490085"
    - "MR3885145"
  point_scheme: 'a quartic elliptic curve \(E\) and four points (the same curve as Sklyanin)'
  centre: '\(\dim \operatorname{Z}(A)_2 = 0\): the two quadrics are imposed as relations, not central'
  code_derive:
    - "alpha = -(beta + gamma)/(1 + beta*gamma)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '-\mathrm{id}'
  nakayama_type: 'scalar'
  nakayama_constant: true
  homological_det: 1
  nakayama_field: 'qq'
s-infinity-twist:
  name: '\(\mathrm{S}_\infty\) twist'
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 2
  hh0: [1, 8, 17]
  hh_qgr: [1, 0, 14, 19]
  point_scheme_dim: 0
  line_scheme_dim: 1
  line_scheme: 'five quartic curves'
  year: 2016
  slug: "s-infinity-twist"
  sortkey: "2016 s-infinity-twist"
  ks_rank: 2
  ks_inj: true
  ks_surj: false
  centre_z2: 1
  centre_z3: 0
  centre_z4: 2
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "β, γ"
      description: >
        With \(α = -(β + γ)/(1 + βγ)\); \(α, β, γ\) nonzero and not \(\pm 1\).
  introduced:
    reference: "MR3490085"
    note: >
      The cocycle twist of \(\mathrm{S}_\infty\) by the Klein four-group, in the same family
      of exotic elliptic algebras studied by Davies and by Chirvasitu–Smith.
  construction: >
    As 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
    \(\mathbb{P}^1 \times \mathbb{P}^1\).
  relations:
    - "(x_1 x_2 - x_2 x_1) - α (x_3 x_4 - x_4 x_3)"
    - "(x_1 x_2 + x_2 x_1) - (x_3 x_4 + x_4 x_3)"
    - "(x_1 x_3 - x_3 x_1) - β (x_4 x_2 - x_2 x_4)"
    - "(x_1 x_3 + x_3 x_1) - (x_4 x_2 + x_2 x_4)"
    - "-x_1^2 + x_2^2 + x_3^2 - x_4^2"
    - "x_2^2 + ((1+α)/(1-β)) x_3^2 - ((1-α)/(1+γ)) x_4^2"
  references:
    - "MR3490085"
    - "MR3885145"
  point_scheme: '20 points'
  centre: '\(\dim \operatorname{Z}(A)_2 = 1\); the central quadric is a smooth \(\mathbb{P}^1 \times \mathbb{P}^1\)'
  code_derive:
    - "alpha = -(beta + gamma)/(1 + beta*gamma)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '-\mathrm{id}'
  nakayama_type: 'scalar'
  nakayama_constant: true
  homological_det: 1
  nakayama_field: 'qq'
shelton-tingey:
  name: "Shelton–Tingey"
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 0
  hh0: [1, 1, 7]
  hh_qgr: [1, 0, 16, 21]
  year: 2001
  point_scheme_dim: 0
  line_scheme_dim: 1
  line_scheme: 'a quartic curve, four cubic curves and two conics'
  point_scheme: '20 points'
  slug: "shelton-tingey"
  sortkey: "2001 shelton-tingey"
  ks_rank: 0
  ks_inj: true
  ks_surj: false
  centre_z2: 0
  centre_z3: 0
  centre_z4: 2
  normal_1: -1
  normal_2: 0
  parameters:
    - symbol: "i"
      description: "a primitive 4th root of unity (\\(i^2 = -1\\)) in the base field; no free parameters."
  introduced:
    reference: "MR1843325"
    note: >
      Example 3.1 of Shelton–Tingey, from their general construction of
      Artin–Schelter regular algebras out of Koszul algebras.
  construction: >
    A 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 \(i\).
  relations:
    - "x_3 x_1 - x_1 x_3 + x_2^2"
    - "i x_4 x_1 + x_1 x_4"
    - "x_4 x_2 - x_2 x_4 + x_3^2"
    - "i x_3 x_2 + x_2 x_3"
    - "x_1^2 - x_3^2"
    - "x_2^2 - x_4^2"
  references:
    - "MR1843325"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: true
  nakayama: '\mathrm{id}'
  nakayama_type: 'identity'
  homological_det: 1
  nakayama_field: 'gf'
skew:
  name: "skew"
  kind: "named"
  pym: true
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 6
  hh0: [4, 6, 4]
  hh_qgr: [1, 3, 3, 5]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'four planes'
  year: 1990
  slug: "skew"
  sortkey: "1990 skew"
  ks_rank: 6
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "q_{ij}"
      description: 'one nonzero scalar for each pair \(i < j\) (six in total).'
  introduced:
    reference: "MR3527537"
    note: >
      The generic quantum (skew) polynomial ring; its point variety was determined
      by Belmans–De Laet–Le Bruyn. It also occurs as the four-generator graded
      skew Clifford algebra.
  construction: >
    The 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.
  relations:
    - "x_1 x_2 - q_{12} x_2 x_1"
    - "x_1 x_3 - q_{13} x_3 x_1"
    - "x_1 x_4 - q_{14} x_4 x_1"
    - "x_2 x_3 - q_{23} x_3 x_2"
    - "x_2 x_4 - q_{24} x_4 x_2"
    - "x_3 x_4 - q_{34} x_4 x_3"
  references:
    - "MR3527537"
    - "MR1086882"
  point_scheme: 'six lines, the tetrahedron skeleton \(K_4\)'
  centre: 'trivial in low degrees'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(q_{12}q_{13}q_{14},\tfrac{q_{23}q_{24}}{q_{12}},\tfrac{q_{34}}{q_{13}q_{23}},\tfrac{1}{q_{14}q_{24}q_{34}}\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
sklyanin:
  name: "Sklyanin"
  kind: "named"
  pym: true
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 2
  hh0: [1, 2, 9]
  hh_qgr: [1, 0, 2, 7]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'the surface of secant lines to the point-scheme quartic elliptic curve \(E\) — an elliptic ruled surface of degree 8'
  year: 1982
  slug: "sklyanin"
  sortkey: "1982 sklyanin"
  ks_rank: 2
  ks_inj: true
  ks_surj: true
  centre_z2: 2
  centre_z3: 0
  centre_z4: 3
  normal_1: -1
  normal_2: 1
  parameters:
    - symbol: "β, γ"
      description: >
        Two scalar parameters; the third is \(α = -(β + γ)/(1 + βγ)\), so that
        \(α + β + γ + αβγ = 0\). Generic \((β, γ)\) with \(α, β, γ \notin \{0, \pm 1\}\).
  introduced:
    reference: "MR684124"
    note: >
      The 4-dimensional Sklyanin algebra, introduced by Sklyanin (1982/1983) in
      connection with the quantum Yang–Baxter equation and elliptic \(R\)-matrices.
      Its Artin–Schelter regularity was established by Smith–Stafford.
  construction: >
    The "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 \(k[\Omega_1, \Omega_2]\).
  relations:
    - "(x_1 x_2 - x_2 x_1) - α (x_3 x_4 + x_4 x_3)"
    - "(x_1 x_2 + x_2 x_1) - (x_3 x_4 - x_4 x_3)"
    - "(x_1 x_3 - x_3 x_1) - β (x_4 x_2 + x_2 x_4)"
    - "(x_1 x_3 + x_3 x_1) - (x_4 x_2 - x_2 x_4)"
    - "(x_1 x_4 - x_4 x_1) - γ (x_2 x_3 + x_3 x_2)"
    - "(x_1 x_4 + x_4 x_1) - (x_2 x_3 - x_3 x_2)"
  references:
    - "MR684124"
    - "MR1175941"
  point_scheme: 'a quartic elliptic curve \(E\) and four points'
  centre: '\(k[\Omega_1, \Omega_2]\) (GK-dimension \(2\)); not module-finite over the centre when \(σ\) has infinite order'
  code_derive:
    - "alpha = -(beta + gamma)/(1 + beta*gamma)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: true
  nakayama: '\mathrm{id}'
  nakayama_type: 'identity'
  nakayama_constant: true
  homological_det: 1
  nakayama_field: 'qq'
sklyanin-twist:
  name: "Sklyanin twist"
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 2
  hh0: [1, 8, 21]
  hh_qgr: [1, 0, 14, 19]
  point_scheme_dim: 0
  line_scheme_dim: 1
  line_scheme: 'three quartic curves and four conics'
  year: 2016
  slug: "sklyanin-twist"
  sortkey: "2016 sklyanin-twist"
  ks_rank: 2
  ks_inj: true
  ks_surj: false
  centre_z2: 2
  centre_z3: 0
  centre_z4: 3
  normal_1: -1
  normal_2: 1
  parameters:
    - symbol: "β, γ"
      description: >
        As for the Sklyanin algebra, with \(α = -(β + γ)/(1 + βγ)\).
  introduced:
    reference: "MR3490085"
    note: >
      The cocycle twist of the 4-dimensional Sklyanin algebra by the Klein
      four-group \(G = (\mathbb{Z}/2)^2\), studied by Davies.
  construction: >
    A 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.
  relations:
    - "(x_1 x_2 - x_2 x_1) - α (x_3 x_4 - x_4 x_3)"
    - "(x_1 x_2 + x_2 x_1) - (x_3 x_4 + x_4 x_3)"
    - "(x_1 x_3 - x_3 x_1) - β (x_4 x_2 - x_2 x_4)"
    - "(x_1 x_3 + x_3 x_1) - (x_4 x_2 + x_2 x_4)"
    - "(x_1 x_4 - x_4 x_1) + γ (x_2 x_3 - x_3 x_2)"
    - "(x_1 x_4 + x_4 x_1) + (x_2 x_3 + x_3 x_2)"
  references:
    - "MR3490085"
    - "MR3885145"
  point_scheme: '20 points'
  centre: 'still \(k[\Omega_1, \Omega_2]\) (GK-dimension \(2\)); not a cocycle-twist invariant'
  code_derive:
    - "alpha = -(beta + gamma)/(1 + beta*gamma)"

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: true
  nakayama: '\mathrm{id}'
  nakayama_type: 'identity'
  nakayama_constant: true
  homological_det: 1
  nakayama_field: 'qq'
vancliff:
  name: "Vancliff"
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 3
  hh0: [3, 4, 3]
  hh_qgr: [1, 2, 4, 7]
  point_scheme_dim: 2
  line_scheme_dim: 3
  line_scheme: 'a quadric threefold and two conics'
  year: 1994
  slug: "vancliff"
  sortkey: "1994 vancliff"
  ks_rank: 3
  ks_inj: true
  ks_surj: false
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "α, β, λ"
      description: 'Scalars with \(λ \neq αβ\).'
  introduced:
    reference: "MR1272579"
    note: >
      Vancliff's quadratic algebra associated with the union of a quadric and a
      line in \(\mathbb{P}^3\).
  construction: >
    Built 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 \(x_1 x_4\).
  relations:
    - "x_2 x_1 - α x_1 x_2"
    - "x_3 x_1 - λ x_1 x_3"
    - "x_4 x_1 - α λ x_1 x_4"
    - "x_4 x_3 - α x_3 x_4"
    - "x_4 x_2 - λ x_2 x_4"
    - "x_3 x_2 - β x_2 x_3 - (α β - λ) x_1 x_4"
  references:
    - "MR1272579"
  point_scheme: 'a line and a quadric surface'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(\tfrac{1}{a^{2}l^{2}},\tfrac{a}{bl},\tfrac{bl}{a},a^{2}l^{2}\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'
vancliff-twist:
  name: "Vancliff twist"
  kind: "named"
  generators: ["x_1", "x_2", "x_3", "x_4"]
  quadratic: true
  num_parameters: 3
  hh0: [3, 4, 3]
  hh_qgr: [1, 2, 4, 7]
  point_scheme_dim: 1
  line_scheme_dim: 2
  line_scheme: 'a quadric surface, two planes and two conics'
  year: 1994
  slug: "vancliff-twist"
  sortkey: "1994 vancliff-twist"
  ks_rank: 3
  ks_inj: true
  ks_surj: false
  centre_z2: 0
  centre_z3: 0
  centre_z4: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "α, β, λ"
      description: 'Scalars with \(λ \neq αβ\).'
  introduced:
    reference: "MR1272579"
    note: >
      A twist of Vancliff's quadric-and-line algebra, appearing among the
      components classified in arXiv:2511.08390.
  construction: >
    Obtained 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.
  relations:
    - "x_2 x_1 - α x_1 x_2"
    - "x_3 x_1 - λ x_1 x_3"
    - "x_4 x_1 - α λ x_1 x_4"
    - "x_4 x_3 + α x_3 x_4"
    - "x_4 x_2 + λ x_2 x_4"
    - "x_3 x_2 + β x_2 x_3 - (α β - λ) x_1 x_4"
  references:
    - "MR1272579"
  point_scheme: 'five lines (\(K_4\) minus one edge)'

  # Nakayama automorphism (computed; see code/nakayama.m2)
  calabi_yau: false
  nakayama: '\operatorname{diag}\!\left(\tfrac{1}{a^{2}l^{2}},\tfrac{a}{bl},\tfrac{bl}{a},a^{2}l^{2}\right)'
  nakayama_type: 'diagonal'
  nakayama_constant: false
  homological_det: 1
  nakayama_field: 'qq'