double Ore P
- Relations
- \(x_2 x_1 + x_1 x_2\)
- \(x_4 x_3 + x_3 x_4\)
- \(x_3 x_1 + h (- x_1 x_4 - f x_2 x_4)\)
- \(x_3 x_2 + h (- x_1 x_4 - x_2 x_4)\)
- \(x_4 x_1 + h (- x_1 x_3 + f x_2 x_3)\)
- \(x_4 x_2 + h (x_1 x_3 - x_2 x_3)\)
- Parameters
- \(f\) — scalar parameter
- \(h\) — deformation parameter (generic scalar)
- Point scheme
- a union of 2 lines (dimension \(1\)).
- Line scheme
- a curve of degree 20 (dimension \(1\)).
- 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 \(A\)
- \(\mathrm{HH}^\bullet_0(A) = (1, 2, 2, 2)\)
- Hochschild cohomology of \(\operatorname{qgr} A\)
- \(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 1, 15, 19)\)
- Kodaira–Spencer
- rank \(2\)
injective: yes
surjective: yes - Nakayama automorphism
- \(\nu = \operatorname{diag}\!\left(\tfrac{1}{(f-1)h^{2}},\tfrac{1}{(f-1)h^{2}},(f-1)h^{2},(f-1)h^{2}\right)\).
Depends on the parameters.
Homological determinant \(\det \nu = 1\).
Computed exactly over \(\mathbb{Q}(\text{params})\). - Notes
- Constraint: \(f \neq 1\).
- Introduced
- 2009, MR2529094. Family P 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[f, h]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
x2*x1 + x1*x2,
x4*x3 + x3*x4,
x3*x1 + h*(-x1*x4 - f*x2*x4),
x3*x2 + h*(-x1*x4 - x2*x4),
x4*x1 + h*(-x1*x3 + f*x2*x3),
x4*x2 + h*(x1*x3 - x2*x3)
};
B = A/I;PolyRing := FunctionField(Rationals, ["f", "h"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
f := indets[1];;
h := indets[2];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
x2*x1 + x1*x2,
x4*x3 + x3*x4,
x3*x1 + h*(-x1*x4 - f*x2*x4),
x3*x2 + h*(-x1*x4 - x2*x4),
x4*x1 + h*(-x1*x3 + f*x2*x3),
x4*x2 + h*(x1*x3 - x2*x3)
];;
A := kQ / rels;;// untested (Magma not available here)
K<f, h> := RationalFunctionField(Rationals(), 2);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
x2*x1 + x1*x2,
x4*x3 + x3*x4,
x3*x1 + h*(-x1*x4 - f*x2*x4),
x3*x2 + h*(-x1*x4 - x2*x4),
x4*x1 + h*(-x1*x3 + f*x2*x3),
x4*x2 + h*(x1*x3 - x2*x3)
];
A := quo< F | rels >;