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\(\mathrm{S}_\infty\)

\(\mathrm{S}_\infty\) keeps the first four (commutator/anticommutator) Sklyanin relations but replaces the last two by the quadrics \(\Omega_1 = -x_1^2 + x_2^2 + x_3^2 + x_4^2\) and \(\Omega_2 = x_2^2 + \tfrac{1+α}{1-β} x_3^2 + \tfrac{1-α}{1+γ} x_4^2\), 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 \(1\)).

Line scheme

the surface of secant lines to the point-scheme quartic elliptic curve \(E\) (the same curve as Sklyanin) — an elliptic ruled surface of degree 8 (dimension \(2\)).

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 of \(A\)

\(\mathrm{HH}^\bullet_0(A) = (1, 1, 2, 5)\)

Hochschild cohomology of \(\operatorname{qgr} A\)

\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 0, 2, 7)\)

Kodaira–Spencer

rank \(2\)
injective: yes
surjective: yes

Nakayama automorphism

\(\nu = -\mathrm{id}\).
Constant over the parameter space.
Homological determinant \(\det \nu = 1\).
Computed exactly over \(\mathbb{Q}(\text{params})\).

Introduced

2016, MR3490085. One of the algebras \(\mathrm{S}_\infty\) arising in Davies' study of cocycle twists of the 4-dimensional Sklyanin algebra; an exotic elliptic algebra in the sense of Chirvasitu–Smith. The subscript \(\infty\) marks it as the degenerate limit of Davies' one-parameter family \(\mathrm{S}_d\) of twisted Sklyanin algebras (the \(d \to \infty\) member), where the two central quadrics of the Sklyanin algebra are instead imposed as defining relations.

References

Code

The presentation, ready to paste into a computer algebra system:

• Macaulay2
• GAP
• Magma

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 >;