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Sklyanin

The "elliptic" noncommutative \(\mathbb{P}^3\). The algebra is built from an elliptic curve \(E\) together with a translation automorphism \(σ\): the six quadratic relations encode theta-function identities on \(E\), and the point scheme is \(E\) itself (a quartic) together with four \(σ\)-fixed points. It carries a pencil of central quadrics \(k[\Omega_1, \Omega_2]\).

Relations

• \((x_1 x_2 - x_2 x_1) - α (x_3 x_4 + x_4 x_3)\)
• \((x_1 x_2 + x_2 x_1) - (x_3 x_4 - x_4 x_3)\)
• \((x_1 x_3 - x_3 x_1) - β (x_4 x_2 + x_2 x_4)\)
• \((x_1 x_3 + x_3 x_1) - (x_4 x_2 - x_2 x_4)\)
• \((x_1 x_4 - x_4 x_1) - γ (x_2 x_3 + x_3 x_2)\)
• \((x_1 x_4 + x_4 x_1) - (x_2 x_3 - x_3 x_2)\)

Parameters

• \(β, γ\) — Two scalar parameters; the third is \(α = -(β + γ)/(1 + βγ)\), so that \(α + β + γ + αβγ = 0\). Generic \((β, γ)\) with \(α, β, γ \notin \{0, \pm 1\}\).

Point scheme

a quartic elliptic curve \(E\) and four points (dimension \(1\)).

Line scheme

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

Centre

\(k[\Omega_1, \Omega_2]\) (GK-dimension \(2\)); not module-finite over the centre when \(σ\) has infinite order.

Normal elements

normal locus dimension \(-1\) (degree 1), \(1\) (degree 2).

Hochschild cohomology of \(A\)

\(\mathrm{HH}^\bullet_0(A) = (1, 1, 2, 9)\)

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

\(\nu = \mathrm{id}\). The algebra is Calabi–Yau.
Constant over the parameter space.
Homological determinant \(\det \nu = 1\).
Computed exactly over \(\mathbb{Q}(\text{params})\).

Introduced

1982, MR684124. The 4-dimensional Sklyanin algebra, introduced by Sklyanin (1982/1983) in connection with the quantum Yang–Baxter equation and elliptic \(R\)-matrices. Its Artin–Schelter regularity was established by Smith–Stafford.

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*x4 - x4*x1) - gamma*(x2*x3 + x3*x2),
  (x1*x4 + x4*x1) - (x2*x3 - x3*x2)
};
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*x4 - x4*x1) - gamma*(x2*x3 + x3*x2),
  (x1*x4 + x4*x1) - (x2*x3 - x3*x2)
];;
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*x4 - x4*x1) - gamma*(x2*x3 + x3*x2),
  (x1*x4 + x4*x1) - (x2*x3 - x3*x2)
];
A := quo< F | rels >;