Cassidy–Vancliff 3
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
- \(x_1 x_3 + u_{13} x_3 x_1\)
- \(x_1 x_4 + u_{14} x_4 x_1\)
- \(x_3 x_4 + u_{34} x_4 x_3\)
- \(x_4^2 - x_2^2\)
- \(x_2 x_3 + x_3 x_2 + x_4^2\)
- \(x_2 x_4 + u_{24} x_4 x_2 + x_1^2\)
- Parameters
- \(u_{13}, u_{14}, u_{24}, u_{34}\) — Four scalars subject to a constraint variety for regularity (\(u_{34}^2 = 1\), \(u_{34} = u_{24}\), \(u_{14}^2 = u_{13}^2\)).
- Point scheme
- 20 points (dimension \(0\)).
- Centre
- \(\dim \operatorname{Z}(A)_3 = 1\): a central cubic.
- Hochschild cohomology
- \(\mathrm{HH}^\bullet_0(A)\) \(= (1, 1, 9, 7)\)
- Kodaira–Spencer
- rank \(4\); injective: yes; 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[u13, u14, u24, u34]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
x1*x3 + u13*x3*x1,
x1*x4 + u14*x4*x1,
x3*x4 + u34*x4*x3,
x4^2 - x2^2,
x2*x3 + x3*x2 + x4^2,
x2*x4 + u24*x4*x2 + x1^2
};
B = A/I;PolyRing := FunctionField(Rationals, ["u13", "u14", "u24", "u34"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
u13 := indets[1];;
u14 := indets[2];;
u24 := indets[3];;
u34 := indets[4];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
x1*x3 + u13*x3*x1,
x1*x4 + u14*x4*x1,
x3*x4 + u34*x4*x3,
x4^2 - x2^2,
x2*x3 + x3*x2 + x4^2,
x2*x4 + u24*x4*x2 + x1^2
];;
A := kQ / rels;;// untested (Magma not available here)
K<u13, u14, u24, u34> := RationalFunctionField(Rationals(), 4);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
x1*x3 + u13*x3*x1,
x1*x4 + u14*x4*x1,
x3*x4 + u34*x4*x3,
x4^2 - x2^2,
x2*x3 + x3*x2 + x4^2,
x2*x4 + u24*x4*x2 + x1^2
];
A := quo< F | rels >;