Cassidy–Vancliff 2
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
- \(x_3 x_1 + x_1 x_3 - β_2 x_2^2\)
- \(x_4 x_1 + x_1 x_4 - α_2 x_3^2\)
- \(x_2 x_3 - x_3 x_2\)
- \(x_4^2 - x_2^2\)
- \(x_4 x_2 + x_2 x_4 - x_3^2\)
- \(α_1 x_3^2 + β_1 x_2^2 - x_1^2\)
- Parameters
- \(α_1, α_2, β_1, β_2\) — Four scalars.
- Point scheme
- 20 points (dimension \(0\)).
- Centre
- \(\dim \operatorname{Z}(A)_2 = 2\).
- Normal elements
- normal locus dimension \(-1\) (degree 1), \(1\) (degree 2).
- Hochschild cohomology
- \(\mathrm{HH}^\bullet_0(A)\) \(= (1, 1, 9, 19)\)
\(\mathrm{HH}^\bullet(\operatorname{qgr} A)\) \(= (1, 0, 16, 21)\) - Kodaira–Spencer
- rank \(3\); injective: no; surjective: no.
- Introduced
- 2010, MR2580455 . A graded skew Clifford / complete-intersection example of Cassidy–Vancliff.
- References
- MR2580455 , arXiv:2511.08390
Code
The presentation, ready to paste into a computer algebra system:
needsPackage "AssociativeAlgebras"
K = frac(QQ[alpha1, alpha2, beta1, beta2]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
x3*x1 + x1*x3 - beta2*x2^2,
x4*x1 + x1*x4 - alpha2*x3^2,
x2*x3 - x3*x2,
x4^2 - x2^2,
x4*x2 + x2*x4 - x3^2,
alpha1*x3^2 + beta1*x2^2 - x1^2
};
B = A/I;PolyRing := FunctionField(Rationals, ["alpha1", "alpha2", "beta1", "beta2"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
alpha1 := indets[1];;
alpha2 := indets[2];;
beta1 := indets[3];;
beta2 := indets[4];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
x3*x1 + x1*x3 - beta2*x2^2,
x4*x1 + x1*x4 - alpha2*x3^2,
x2*x3 - x3*x2,
x4^2 - x2^2,
x4*x2 + x2*x4 - x3^2,
alpha1*x3^2 + beta1*x2^2 - x1^2
];;
A := kQ / rels;;// untested (Magma not available here)
K<alpha1, alpha2, beta1, beta2> := RationalFunctionField(Rationals(), 4);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
x3*x1 + x1*x3 - beta2*x2^2,
x4*x1 + x1*x4 - alpha2*x3^2,
x2*x3 - x3*x2,
x4^2 - x2^2,
x4*x2 + x2*x4 - x3^2,
alpha1*x3^2 + beta1*x2^2 - x1^2
];
A := quo< F | rels >;