double Ore C
- Relations
- \(x_4 x_3 - p x_3 x_4\)
- \(x_2 x_1 - p x_1 x_2\)
- \(x_3 x_1 + h (x_1 x_3 - p^2 x_2 x_3 - x_1 x_4 + p x_2 x_4)\)
- \(x_3 x_2 + h (p x_1 x_3 - x_2 x_3 - x_1 x_4 + p x_2 x_4)\)
- \(x_4 x_1 + h (p x_1 x_3 + 2 p^2 x_2 x_3 - p x_1 x_4 + p x_2 x_4)\)
- \(x_4 x_2 + h (p x_1 x_3 - p^2 x_2 x_3 - x_1 x_4 + x_2 x_4)\)
- Parameters
- \(h\) — deformation parameter (generic scalar)
- \(p\) — \(p\) is a primitive cube root of unity (\(p^2 + p + 1 = 0\))
- Point scheme
- a regulus of 5 lines (dimension \(1\)).
- Line scheme
- a quadric surface, a quartic curve, two conics and four lines (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 \(-1\) (degree 1), \(-1\) (degree 2).
- Hochschild cohomology of \(A\)
- \(\mathrm{HH}^\bullet_0(A) = (1, 2, 1, 0)\)
- Hochschild cohomology of \(\operatorname{qgr} A\)
- \(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 1, 0, 4)\)
- Kodaira–Spencer
- rank \(1\)
injective: yes
surjective: yes - Nakayama automorphism
- \(\nu = \operatorname{diag}\!\left(-\tfrac{\omega}{3}h^{-2},-\tfrac{\omega}{3}h^{-2},-3\omega^{2}h^{2},-3\omega^{2}h^{2}\right)\).
Depends on the parameters.
Homological determinant \(\det \nu = 1\).
Computed exactly over \(\mathbb{F}_{60013}(\text{params})\). - Introduced
- 2009, MR2529094. Family C among the double extension regular algebras of type (14641) classified by Zhang–Zhang.
References
Code
The presentation, ready to paste into a computer algebra system:
needsPackage "AssociativeAlgebras"
K = frac(QQ[h, p]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
x4*x3 - p*x3*x4,
x2*x1 - p*x1*x2,
x3*x1 + h*(x1*x3 - p^2*x2*x3 - x1*x4 + p*x2*x4),
x3*x2 + h*(p*x1*x3 - x2*x3 - x1*x4 + p*x2*x4),
x4*x1 + h*(p*x1*x3 + 2*p^2*x2*x3 - p*x1*x4 + p*x2*x4),
x4*x2 + h*(p*x1*x3 - p^2*x2*x3 - x1*x4 + x2*x4)
};
B = A/I;PolyRing := FunctionField(Rationals, ["h", "p"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
h := indets[1];;
p := indets[2];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
x4*x3 - p*x3*x4,
x2*x1 - p*x1*x2,
x3*x1 + h*(x1*x3 - p^2*x2*x3 - x1*x4 + p*x2*x4),
x3*x2 + h*(p*x1*x3 - x2*x3 - x1*x4 + p*x2*x4),
x4*x1 + h*(p*x1*x3 + 2*p^2*x2*x3 - p*x1*x4 + p*x2*x4),
x4*x2 + h*(p*x1*x3 - p^2*x2*x3 - x1*x4 + x2*x4)
];;
A := kQ / rels;;// untested (Magma not available here)
K<h, p> := RationalFunctionField(Rationals(), 2);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
x4*x3 - p*x3*x4,
x2*x1 - p*x1*x2,
x3*x1 + h*(x1*x3 - p^2*x2*x3 - x1*x4 + p*x2*x4),
x3*x2 + h*(p*x1*x3 - x2*x3 - x1*x4 + p*x2*x4),
x4*x1 + h*(p*x1*x3 + 2*p^2*x2*x3 - p*x1*x4 + p*x2*x4),
x4*x2 + h*(p*x1*x3 - p^2*x2*x3 - x1*x4 + x2*x4)
];
A := quo< F | rels >;