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

cassidy-vancliff-1:
  name: "Cassidy–Vancliff 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
  year: 2010
  slug: "cassidy-vancliff-1"
  sortkey: "2010 cassidy-vancliff-1"
  ks_rank: 1
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  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: >
      An example of a graded skew Clifford algebra of Cassidy–Vancliff.
  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"
    - "2511.08390"
  point_scheme: '20 points'
cassidy-vancliff-2:
  name: "Cassidy–Vancliff 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
  year: 2010
  slug: "cassidy-vancliff-2"
  sortkey: "2010 cassidy-vancliff-2"
  ks_rank: 3
  ks_inj: false
  ks_surj: false
  centre_z2: 2
  centre_z3: 0
  normal_1: -1
  normal_2: 1
  parameters:
    - symbol: "α_1, α_2, β_1, β_2"
      description: "Four scalars."
  introduced:
    reference: "MR2580455"
    note: >
      A graded skew Clifford / complete-intersection example of Cassidy–Vancliff.
  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"
    - "2511.08390"
  point_scheme: '20 points'
  centre: '\(\dim \operatorname{Z}(A)_2 = 2\)'
cassidy-vancliff-3:
  name: "Cassidy–Vancliff 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
  year: 2010
  slug: "cassidy-vancliff-3"
  sortkey: "2010 cassidy-vancliff-3"
  ks_rank: 4
  ks_inj: true
  ks_surj: false
  centre_z2: 0
  centre_z3: 1
  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: >
      A graded skew Clifford / complete-intersection example of Cassidy–Vancliff.
  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"
    - "2511.08390"
  point_scheme: '20 points'
  centre: '\(\dim \operatorname{Z}(A)_3 = 1\): a central cubic'
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
  year: 1995
  slug: "clifford"
  sortkey: "1995 clifford"
  ks_rank: 9
  ks_inj: false
  ks_surj: true
  centre_z2: 4
  centre_z3: 0
  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"
    - "2511.08390"
  point_scheme: '20 points'
  centre: '\(\dim \operatorname{Z}(A)_2 = 4\) (the four squares are central)'
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]
  year: "?"
  point_scheme_dim: 1
  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
  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:
    - "2511.08390"
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]
  year: "?"
  point_scheme_dim: 1
  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
  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:
    - "2511.08390"
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]
  year: "?"
  point_scheme_dim: 1
  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
  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:
    - "2511.08390"
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]
  year: "?"
  point_scheme_dim: 1
  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
  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:
    - "2511.08390"
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]
  year: "?"
  point_scheme_dim: 1
  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
  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:
    - "2511.08390"
l112sigma:
  name: "L(1,1,2) σ"
  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]
  point_scheme_dim: 1
  year: 2015
  slug: "l112sigma"
  sortkey: "2015 l112sigma"
  ks_rank: 4
  ks_inj: false
  ks_surj: true
  centre_z2: 0
  centre_z3: 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: "MR3366864"
    note: >
      The twisted version of Pym's \(L(1,1,2)\) algebra, a quantization of one of
      the quadratic Poisson structures on \(\mathbb{P}^3\).
  construction: >
    A quantization of the Poisson structure of type \(L(1,1,2)\): \(x_1, x_2\)
    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 parameters
    \(α, β\) implement a further diagonal (\(σ\)) twist.
  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:
    - "MR3366864"
    - "2511.08390"
  point_scheme: 'a line, two conics and two points'
lebruyn:
  name: "central extension of Sklyanin"
  kind: "named"
  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
  year: 1996
  slug: "lebruyn"
  sortkey: "1996 lebruyn"
  ks_rank: 7
  ks_inj: false
  ks_surj: true
  centre_z2: 1
  centre_z3: 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"
    - "2511.08390"
  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'
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
  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
  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)"
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
  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
  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)"
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
  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
  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)"
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
  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
  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)"
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
  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
  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)"
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
  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
  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)"
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
  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
  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)"
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
  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
  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"
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
  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
  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)"
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
  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
  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)"
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
  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
  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)"
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
  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
  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)"
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
  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
  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)"
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
  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
  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: \(f^2 \neq g^2\).'
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
  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
  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: \(f \neq 1\).'
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
  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
  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: \(f \neq 1\).'
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
  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
  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)"
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
  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
  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)"
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
  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
  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)"
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
  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
  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)"
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
  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
  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)"
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
  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
  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"
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
  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
  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."
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
  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
  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)"
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
  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
  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)"
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
  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
  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: \(f(1+f) \neq 0\).'
pym:
  name: "Pym S(2,3)"
  kind: "named"
  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
  year: 2015
  slug: "pym"
  sortkey: "2015 pym"
  ks_rank: 9
  ks_inj: false
  ks_surj: true
  centre_z2: 0
  centre_z3: 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"
    - "2511.08390"
  point_scheme: 'three lines and a rational normal quartic'
r3a:
  name: "R(3,a)"
  kind: "named"
  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
  year: 2016
  slug: "r3a"
  sortkey: "2016 r3a"
  ks_rank: 1
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  normal_1: 0
  normal_2: 0
  parameters:
    - symbol: "a"
      description: "A single scalar parameter."
  introduced:
    reference: "MR3901632"
    note: >
      A member of the additive (\(\mathbb{G}_a\)) 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 \(\mathbb{G}_a\)-symmetry. Its point modules are parametrised by a bouquet of
    rational normal curves (a line, a conic and a twisted cubic).
  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"
    - "2511.08390"
  point_scheme: 'a line, a conic and a twisted cubic (a bouquet of rational normal curves)'
s-infinity:
  name: '\(S_\infty\)'
  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
  year: 2016
  slug: "s-infinity"
  sortkey: "2016 s-infinity"
  ks_rank: 2
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 0
  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 degenerate "limit" algebras \(S_\infty\) arising in Davies' study
      of cocycle twists of the 4-dimensional Sklyanin algebra; an exotic elliptic
      algebra in the sense of Chirvasitu–Smith.
  construction: >
    \(S_\infty\) keeps the first four (commutator/anticommutator) Sklyanin
    relations but replaces the last two by the quadrics
    \(Ω_1 = -x_1^2 + x_2^2 + x_3^2 + x_4^2\) and
    \(Ω_2 = x_2^2 + \tfrac{1+α}{1-β} x_3^2 + \tfrac{1-α}{1+γ} x_4^2\), 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"
    - "2511.08390"
  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)"
s-infinity-twist:
  name: '\(S_\infty\) 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
  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
  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 \(S_\infty\) by the Klein four-group, in the same family
      of exotic elliptic algebras studied by Davies and by Chirvasitu–Smith.
  construction: >
    As for \(S_\infty\), 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"
    - "2511.08390"
  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)"
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]
  year: 2001
  point_scheme_dim: 0
  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
  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"
    - "2511.08390"
skew:
  name: "skew"
  kind: "named"
  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
  year: 1990
  slug: "skew"
  sortkey: "1990 skew"
  ks_rank: 6
  ks_inj: true
  ks_surj: true
  centre_z2: 0
  centre_z3: 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"
    - "2511.08390"
  point_scheme: 'six lines, the tetrahedron skeleton \(K_4\)'
  centre: 'trivial in low degrees'
sklyanin:
  name: "Sklyanin"
  kind: "named"
  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
  year: 1982
  slug: "sklyanin"
  sortkey: "1982 sklyanin"
  ks_rank: 2
  ks_inj: true
  ks_surj: true
  centre_z2: 2
  centre_z3: 0
  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 \(k[Ω_1, Ω_2]\).
  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"
    - "2511.08390"
  point_scheme: 'a quartic elliptic curve \(E\) and four points'
  centre: '\(k[Ω_1, Ω_2]\) (GK-dimension \(2\)); not module-finite over the centre when \(σ\) has infinite order'
  code_derive:
    - "alpha = -(beta + gamma)/(1 + beta*gamma)"
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
  year: 2016
  slug: "sklyanin-twist"
  sortkey: "2016 sklyanin-twist"
  ks_rank: 2
  ks_inj: true
  ks_surj: false
  centre_z2: 2
  centre_z3: 0
  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"
    - "2511.08390"
  point_scheme: '20 points'
  centre: 'still \(k[Ω_1, Ω_2]\) (GK-dimension \(2\)); not a cocycle-twist invariant'
  code_derive:
    - "alpha = -(beta + gamma)/(1 + beta*gamma)"
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
  year: 1994
  slug: "vancliff"
  sortkey: "1994 vancliff"
  ks_rank: 3
  ks_inj: true
  ks_surj: false
  centre_z2: 0
  centre_z3: 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"
    - "2511.08390"
  point_scheme: 'a line and a quadric surface'
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
  year: 1994
  slug: "vancliff-twist"
  sortkey: "1994 vancliff-twist"
  ks_rank: 3
  ks_inj: true
  ks_surj: false
  centre_z2: 0
  centre_z3: 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"
    - "2511.08390"
  point_scheme: 'five lines (\(K_4\) minus one edge)'