\(S_\infty\) twist
As for \(S_\infty\), but the surviving Sklyanin relations are twisted (commutators exchanged with anticommutators) and the signs in the two quadrics change. The twist trades the elliptic point scheme for 20 points and produces a single central quadric whose symbol is the smooth hyperbolic form \(\mathbb{P}^1 \times \mathbb{P}^1\).
- 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
- 20 points (dimension \(0\)).
- Centre
- \(\dim \operatorname{Z}(A)_2 = 1\); the central quadric is a smooth \(\mathbb{P}^1 \times \mathbb{P}^1\).
- Normal elements
- normal locus dimension \(-1\) (degree 1), \(0\) (degree 2).
- Hochschild cohomology
- \(\mathrm{HH}^\bullet_0(A)\) \(= (1, 1, 8, 17)\)
\(\mathrm{HH}^\bullet(\operatorname{qgr} A)\) \(= (1, 0, 14, 19)\) - Kodaira–Spencer
- rank \(2\); injective: yes; surjective: no.
- Introduced
- 2016, MR3490085 . The cocycle twist of \(S_\infty\) by the Klein four-group, in the same family of exotic elliptic algebras studied by Davies and by 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 >;