double Ore J
- Relations
- \(x_2 x_1 - q x_1 x_2\)
- \(x_4 x_3 + x_3 x_4\)
- \(x_3 x_1 + h (- x_2 x_3 - x_2 x_4)\)
- \(x_3 x_2 + h (x_1 x_3 - x_1 x_4)\)
- \(x_4 x_1 + h (- x_2 x_3 + x_2 x_4)\)
- \(x_4 x_2 + h (- x_1 x_3 - x_1 x_4)\)
- Parameters
- \(h\) — deformation parameter (generic scalar)
- \(q\) — \(q\) is a primitive 4th root of unity (\(q^2 = -1\))
- Point scheme
- a union of 6 lines (dimension \(1\)).
- Line scheme
- two quadric surfaces, two planes and two 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), \(0\) (degree 2).
- Hochschild cohomology of \(A\)
- \(\mathrm{HH}^\bullet_0(A) = (1, 2, 1, 0)\)
- Hochschild cohomology of \(\operatorname{qgr} A\)
- \(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 2, 3, 6)\)
- Kodaira–Spencer
- rank \(1\)
injective: yes
surjective: yes - Nakayama automorphism
- \(\nu = x_1\mapsto-\tfrac{i}{2}h^{-2}x_1,\ x_2\mapsto\tfrac{i}{2}h^{-2}x_2,\ x_3\mapsto2i\,h^{2}x_4,\ x_4\mapsto-2i\,h^{2}x_3\).
Depends on the parameters.
Homological determinant \(\det \nu = -1\).
Computed exactly over \(\mathbb{F}_{60013}(\text{params})\). - Introduced
- 2009, MR2529094. Family J 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, q]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
x2*x1 - q*x1*x2,
x4*x3 + x3*x4,
x3*x1 + h*(-x2*x3 - x2*x4),
x3*x2 + h*(x1*x3 - x1*x4),
x4*x1 + h*(-x2*x3 + x2*x4),
x4*x2 + h*(-x1*x3 - x1*x4)
};
B = A/I;PolyRing := FunctionField(Rationals, ["h", "q"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
h := indets[1];;
q := indets[2];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
x2*x1 - q*x1*x2,
x4*x3 + x3*x4,
x3*x1 + h*(-x2*x3 - x2*x4),
x3*x2 + h*(x1*x3 - x1*x4),
x4*x1 + h*(-x2*x3 + x2*x4),
x4*x2 + h*(-x1*x3 - x1*x4)
];;
A := kQ / rels;;// untested (Magma not available here)
K<h, q> := RationalFunctionField(Rationals(), 2);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
x2*x1 - q*x1*x2,
x4*x3 + x3*x4,
x3*x1 + h*(-x2*x3 - x2*x4),
x3*x2 + h*(x1*x3 - x1*x4),
x4*x1 + h*(-x2*x3 + x2*x4),
x4*x2 + h*(-x1*x3 - x1*x4)
];
A := quo< F | rels >;