\(S_\infty\)
\(S_\infty\) keeps the first four (commutator/anticommutator) Sklyanin relations but replaces the last two by the quadrics \(Ω_1 = -x_1^2 + x_2^2 + x_3^2 + x_4^2\) and \(Ω_2 = x_2^2 + \tfrac{1+α}{1-β} x_3^2 + \tfrac{1-α}{1+γ} x_4^2\), imposed as relations. This kills the central quadrics yet keeps the elliptic quartic as point scheme.
- Relations
- \((x_1 x_2 - x_2 x_1) - α (x_3 x_4 + x_4 x_3)\)
- \((x_1 x_2 + x_2 x_1) - (x_3 x_4 - x_4 x_3)\)
- \((x_1 x_3 - x_3 x_1) - β (x_4 x_2 + x_2 x_4)\)
- \((x_1 x_3 + x_3 x_1) - (x_4 x_2 - x_2 x_4)\)
- \(-x_1^2 + x_2^2 + x_3^2 + x_4^2\)
- \(x_2^2 + ((1+α)/(1-β)) x_3^2 + ((1-α)/(1+γ)) x_4^2\)
- Parameters
- \(β, γ\) — With \(α = -(β + γ)/(1 + βγ)\); \(α, β, γ\) nonzero and not \(\pm 1\).
- Point scheme
- a quartic elliptic curve \(E\) and four points (the same curve as Sklyanin) (dimension \(1\)).
- Centre
- \(\dim \operatorname{Z}(A)_2 = 0\): the two quadrics are imposed as relations, not central.
- Normal elements
- normal locus dimension \(-1\) (degree 1), \(0\) (degree 2).
- Hochschild cohomology
- \(\mathrm{HH}^\bullet_0(A)\) \(= (1, 1, 2, 5)\)
\(\mathrm{HH}^\bullet(\operatorname{qgr} A)\) \(= (1, 0, 2, 7)\) - Kodaira–Spencer
- rank \(2\); injective: yes; surjective: yes.
- Introduced
- 2016, MR3490085 . One of the degenerate "limit" algebras \(S_\infty\) arising in Davies' study of cocycle twists of the 4-dimensional Sklyanin algebra; an exotic elliptic algebra in the sense of Chirvasitu–Smith.
- References
- MR3490085 , MR3885145 , arXiv:2511.08390
Code
The presentation, ready to paste into a computer algebra system:
needsPackage "AssociativeAlgebras"
K = frac(QQ[beta, gamma]);
alpha = -(beta + gamma)/(1 + beta*gamma);
A = K<|x1,x2,x3,x4|>;
I = ideal {
(x1*x2 - x2*x1) - alpha*(x3*x4 + x4*x3),
(x1*x2 + x2*x1) - (x3*x4 - x4*x3),
(x1*x3 - x3*x1) - beta*(x4*x2 + x2*x4),
(x1*x3 + x3*x1) - (x4*x2 - x2*x4),
-x1^2 + x2^2 + x3^2 + x4^2,
x2^2 + ((1+alpha)*/*(1-beta))*x3^2 + ((1-alpha)*/*(1+gamma))*x4^2
};
B = A/I;PolyRing := FunctionField(Rationals, ["beta", "gamma"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
beta := indets[1];;
gamma := indets[2];;
alpha := -(beta + gamma)/(1 + beta*gamma);;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
(x1*x2 - x2*x1) - alpha*(x3*x4 + x4*x3),
(x1*x2 + x2*x1) - (x3*x4 - x4*x3),
(x1*x3 - x3*x1) - beta*(x4*x2 + x2*x4),
(x1*x3 + x3*x1) - (x4*x2 - x2*x4),
-x1^2 + x2^2 + x3^2 + x4^2,
x2^2 + ((1+alpha)*/*(1-beta))*x3^2 + ((1-alpha)*/*(1+gamma))*x4^2
];;
A := kQ / rels;;// untested (Magma not available here)
K<beta, gamma> := RationalFunctionField(Rationals(), 2);
alpha := -(beta + gamma)/(1 + beta*gamma);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
(x1*x2 - x2*x1) - alpha*(x3*x4 + x4*x3),
(x1*x2 + x2*x1) - (x3*x4 - x4*x3),
(x1*x3 - x3*x1) - beta*(x4*x2 + x2*x4),
(x1*x3 + x3*x1) - (x4*x2 - x2*x4),
-x1^2 + x2^2 + x3^2 + x4^2,
x2^2 + ((1+alpha)*/*(1-beta))*x3^2 + ((1-alpha)*/*(1+gamma))*x4^2
];
A := quo< F | rels >;