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Vancliff twist

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

• \(x_2 x_1 - α x_1 x_2\)
• \(x_3 x_1 - λ x_1 x_3\)
• \(x_4 x_1 - α λ x_1 x_4\)
• \(x_4 x_3 + α x_3 x_4\)
• \(x_4 x_2 + λ x_2 x_4\)
• \(x_3 x_2 + β x_2 x_3 - (α β - λ) x_1 x_4\)

Parameters

• \(α, β, λ\) — Scalars with \(λ \neq αβ\).

Point scheme

five lines (\(K_4\) minus one edge) (dimension \(1\)).

Line scheme

a quadric surface, two planes and two conics (dimension \(2\)).

Centre

\(\dim \operatorname{Z}(A)_2 = 0\), \(\dim \operatorname{Z}(A)_3 = 0\), \(\dim \operatorname{Z}(A)_4 = 0\).

Normal elements

normal locus dimension \(0\) (degree 1), \(0\) (degree 2).

Hochschild cohomology of \(A\)

\(\mathrm{HH}^\bullet_0(A) = (1, 3, 4, 3)\)

Hochschild cohomology of \(\operatorname{qgr} A\)

\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 2, 4, 7)\)

Kodaira–Spencer

rank \(3\)
injective: yes
surjective: no

Nakayama automorphism

\(\nu = \operatorname{diag}\!\left(\tfrac{1}{a^{2}l^{2}},\tfrac{a}{bl},\tfrac{bl}{a},a^{2}l^{2}\right)\).
Depends on the parameters.
Homological determinant \(\det \nu = 1\).
Computed exactly over \(\mathbb{Q}(\text{params})\).

Introduced

1994, MR1272579. A twist of Vancliff's quadric-and-line algebra, appearing among the components classified in arXiv:2511.08390.

References

Code

The presentation, ready to paste into a computer algebra system:

• Macaulay2
• GAP
• Magma

needsPackage "AssociativeAlgebras"
K = frac(QQ[alpha, beta, lambda]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
  x2*x1 - alpha*x1*x2,
  x3*x1 - lambda*x1*x3,
  x4*x1 - alpha*lambda*x1*x4,
  x4*x3 + alpha*x3*x4,
  x4*x2 + lambda*x2*x4,
  x3*x2 + beta*x2*x3 - (alpha*beta - lambda)*x1*x4
};
B = A/I;

PolyRing := FunctionField(Rationals, ["alpha", "beta", "lambda"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
alpha := indets[1];;
beta := indets[2];;
lambda := indets[3];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
  x2*x1 - alpha*x1*x2,
  x3*x1 - lambda*x1*x3,
  x4*x1 - alpha*lambda*x1*x4,
  x4*x3 + alpha*x3*x4,
  x4*x2 + lambda*x2*x4,
  x3*x2 + beta*x2*x3 - (alpha*beta - lambda)*x1*x4
];;
A := kQ / rels;;

// untested (Magma not available here)
K<alpha, beta, lambda> := RationalFunctionField(Rationals(), 3);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
  x2*x1 - alpha*x1*x2,
  x3*x1 - lambda*x1*x3,
  x4*x1 - alpha*lambda*x1*x4,
  x4*x3 + alpha*x3*x4,
  x4*x2 + lambda*x2*x4,
  x3*x2 + beta*x2*x3 - (alpha*beta - lambda)*x1*x4
];
A := quo< F | rels >;