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\(\mathrm{S}_{d,i}\)

Take the six defining relations \(f_1, \dots, f_6\) of the Sklyanin algebra and, for one chosen index \(i \in \{1, \dots, 6\}\), replace \(f_i\) by the single quadric \(d_1 \Omega_1 + d_2 \Omega_2\), where \(\Omega_1, \Omega_2\) are the two central quadrics of the Sklyanin algebra and \((d_1 : d_2) \in \mathbb{P}^1\). The five remaining Sklyanin relations \(f_j\) (\(j \neq i\)) are kept verbatim. As \(d \to \infty\) (only the quadrics surviving) one recovers \(\mathrm{S}_\infty\).

Relations

• \((x_1 x_2 - x_2 x_1) - α (x_3 x_4 + x_4 x_3)\)
• \((x_1 x_2 + x_2 x_1) - (x_3 x_4 - x_4 x_3)\)
• \((x_1 x_3 - x_3 x_1) - β (x_4 x_2 + x_2 x_4)\)
• \((x_1 x_3 + x_3 x_1) - (x_4 x_2 - x_2 x_4)\)
• \((x_1 x_4 - x_4 x_1) - γ (x_2 x_3 + x_3 x_2)\)
• \((x_1 x_4 + x_4 x_1) - (x_2 x_3 - x_3 x_2)\)
• \(d_1 (-x_1^2 + x_2^2 + x_3^2 + x_4^2) + d_2 (x_2^2 + ((1+α)/(1-β)) x_3^2 + ((1-α)/(1+γ)) x_4^2)\)

Parameters

• \(β, γ\) — With \(α = -(β + γ)/(1 + βγ)\); \(α, β, γ\) nonzero and not \(\pm 1\).
• \(d = (d_1, d_2)\) — A point of \(\mathbb{P}^1\); these are the coefficients of the two central quadrics \(\Omega_1, \Omega_2\) in the replaced relation.

Centre

\(\dim \operatorname{Z}(A)_2 = 1\), \(\dim \operatorname{Z}(A)_3 = 0\), \(\dim \operatorname{Z}(A)_4 = 2\).

Hochschild cohomology of \(A\)

\(\mathrm{HH}^\bullet_0(A) = (1, 1, 3, 9)\)

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

\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 0, 4, 9)\)

Kodaira–Spencer

rank \(3\)
injective: yes
surjective: yes

Notes

This page represents the general construction. For a chosen index \(i \in \{1, \dots, 6\}\), the algebra \(\mathrm{S}_{d,i}\) has the six relations \(f_1, \dots, f_6\) of the Sklyanin algebra (the first six relations above) with \(f_i\) deleted and the quadric \(d_1 \Omega_1 + d_2 \Omega_2\) (the last relation above) imposed in its place — so exactly six relations in total.

Introduced

2016, MR3490085. One of Davies' algebras \(\mathrm{S}_{d,i}\) (equation 6.1.4), a finite member of the one-parameter family of twisted Sklyanin algebras of which \(\mathrm{S}_\infty\) is the \(d \to \infty\) limit.

References

Code

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

• Macaulay2
• GAP
• Magma

needsPackage "AssociativeAlgebras"
K = frac(QQ[beta, d1, d2, 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),
  d1*(-x1^2 + x2^2 + x3^2 + x4^2) + d2*(x2^2 + ((1+alpha)/(1-beta))*x3^2 + ((1-alpha)/(1+gamma))*x4^2)
};
B = A/I;

PolyRing := FunctionField(Rationals, ["beta", "d1", "d2", "gamma"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
beta := indets[1];;
d1 := indets[2];;
d2 := indets[3];;
gamma := indets[4];;
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),
  d1*(-x1^2 + x2^2 + x3^2 + x4^2) + d2*(x2^2 + ((1+alpha)/(1-beta))*x3^2 + ((1-alpha)/(1+gamma))*x4^2)
];;
A := kQ / rels;;

// untested (Magma not available here)
K<beta, d1, d2, gamma> := RationalFunctionField(Rationals(), 4);
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),
  d1*(-x1^2 + x2^2 + x3^2 + x4^2) + d2*(x2^2 + ((1+alpha)/(1-beta))*x3^2 + ((1-alpha)/(1+gamma))*x4^2)
];
A := quo< F | rels >;