central extension of Sklyanin
Start from a 3-dimensional regular algebra on \(x_1, x_2, x_3\) (here of Sklyanin/elliptic type) and adjoin a central generator \(x_4 = z\), deforming the three defining relations by terms linear in \(z\) plus a multiple of \(z^2\). The extension is again regular, of dimension 4, and is the generic way to build a noncommutative \(\mathbb{P}^3\) with a central hyperplane variable.
- Relations
- \(c x_1^2 + a x_2 x_3 + b x_3 x_2 + l_{11} x_1 x_4 + l_{12} x_2 x_4 + l_{13} x_3 x_4 + α_1 x_4^2\)
- \(c x_2^2 + a x_3 x_1 + b x_1 x_3 + l_{12} x_1 x_4 + l_{22} x_2 x_4 + l_{23} x_3 x_4 + α_2 x_4^2\)
- \(c x_3^2 + a x_1 x_2 + b x_2 x_1 + l_{13} x_1 x_4 + l_{23} x_2 x_4 + l_{33} x_3 x_4 + α_3 x_4^2\)
- \(x_1 x_4 - x_4 x_1\)
- \(x_2 x_4 - x_4 x_2\)
- \(x_3 x_4 - x_4 x_3\)
- Parameters
- \(a, b, c\) — Coefficients of the three base relations.
- \(α_1, α_2, α_3, l_{ij}\) — The data of a central element: a symmetric \(3 \times 3\) matrix \((l_{ij})\) and a vector \((α_i)\), describing how the central generator \(x_4\) enters.
- Point scheme
- a twisted cubic and 8 points (dimension \(1\)).
- Centre
- \(\dim \operatorname{Z}(A)_2 = 1\), \(\dim \operatorname{Z}(A)_3 = 2\) (a central quadric and central cubics by construction); \(x_4\) is central.
- Normal elements
- normal locus dimension \(0\) (degree 1), \(0\) (degree 2).
- Hochschild cohomology
- \(\mathrm{HH}^\bullet_0(A)\) \(= (1, 1, 7, 19)\)
\(\mathrm{HH}^\bullet(\operatorname{qgr} A)\) \(= (1, 0, 7, 12)\) - Kodaira–Spencer
- rank \(7\); injective: no; surjective: yes.
- Introduced
- 1996, MR1429334 . A central extension of a 3-dimensional Artin–Schelter regular algebra, in the sense of Le Bruyn–Smith–Van den Bergh.
- References
- MR1429334 , arXiv:2511.08390
Code
The presentation, ready to paste into a computer algebra system:
needsPackage "AssociativeAlgebras"
K = frac(QQ[a, alpha1, alpha2, alpha3, b, c, l11, l12, l13, l22, l23, l33]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
c*x1^2 + a*x2*x3 + b*x3*x2 + l11*x1*x4 + l12*x2*x4 + l13*x3*x4 + alpha1*x4^2,
c*x2^2 + a*x3*x1 + b*x1*x3 + l12*x1*x4 + l22*x2*x4 + l23*x3*x4 + alpha2*x4^2,
c*x3^2 + a*x1*x2 + b*x2*x1 + l13*x1*x4 + l23*x2*x4 + l33*x3*x4 + alpha3*x4^2,
x1*x4 - x4*x1,
x2*x4 - x4*x2,
x3*x4 - x4*x3
};
B = A/I;PolyRing := FunctionField(Rationals, ["a", "alpha1", "alpha2", "alpha3", "b", "c", "l11", "l12", "l13", "l22", "l23", "l33"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
a := indets[1];;
alpha1 := indets[2];;
alpha2 := indets[3];;
alpha3 := indets[4];;
b := indets[5];;
c := indets[6];;
l11 := indets[7];;
l12 := indets[8];;
l13 := indets[9];;
l22 := indets[10];;
l23 := indets[11];;
l33 := indets[12];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
c*x1^2 + a*x2*x3 + b*x3*x2 + l11*x1*x4 + l12*x2*x4 + l13*x3*x4 + alpha1*x4^2,
c*x2^2 + a*x3*x1 + b*x1*x3 + l12*x1*x4 + l22*x2*x4 + l23*x3*x4 + alpha2*x4^2,
c*x3^2 + a*x1*x2 + b*x2*x1 + l13*x1*x4 + l23*x2*x4 + l33*x3*x4 + alpha3*x4^2,
x1*x4 - x4*x1,
x2*x4 - x4*x2,
x3*x4 - x4*x3
];;
A := kQ / rels;;// untested (Magma not available here)
K<a, alpha1, alpha2, alpha3, b, c, l11, l12, l13, l22, l23, l33> := RationalFunctionField(Rationals(), 12);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
c*x1^2 + a*x2*x3 + b*x3*x2 + l11*x1*x4 + l12*x2*x4 + l13*x3*x4 + alpha1*x4^2,
c*x2^2 + a*x3*x1 + b*x1*x3 + l12*x1*x4 + l22*x2*x4 + l23*x3*x4 + alpha2*x4^2,
c*x3^2 + a*x1*x2 + b*x2*x1 + l13*x1*x4 + l23*x2*x4 + l33*x3*x4 + alpha3*x4^2,
x1*x4 - x4*x1,
x2*x4 - x4*x2,
x3*x4 - x4*x3
];
A := quo< F | rels >;