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

As for \(\mathrm{S}_{d,i}\), but starting from the twisted Sklyanin relations \(f'_1, \dots, f'_6\) (commutators exchanged with anticommutators): for one chosen index \(i \in \{1, \dots, 6\}\), replace \(f'_i\) by the single twisted quadric \(d_1 \Omega'_1 + d_2 \Omega'_2\), keeping the five remaining \(f'_j\) (\(j \neq i\)). The signs in the quadrics change as for the \(\mathrm{S}_\infty\) twist.

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 \(\mathbb{P}^1\); the coefficients of the two twisted quadrics \(\Omega'_1, \Omega'_2\) in the replaced relation.

Hochschild cohomology of \(A\)

\(\mathrm{HH}^\bullet_0(A) = (1, 1, 8, 17)\)

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

\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 0, 14, 19)\)

Kodaira–Spencer

rank \(3\)
injective: yes
surjective: no

Notes

This page represents the general construction. For a chosen index \(i \in \{1, \dots, 6\}\), the algebra has the six twisted Sklyanin relations \(f'_1, \dots, f'_6\) (the first six above) with \(f'_i\) deleted and the twisted quadric \(d_1 \Omega'_1 + d_2 \Omega'_2\) (the last relation above) imposed in its place — so exactly six relations in total. For AS-regularity one excludes the degenerate \((d_1, d_2)\) noted in the source (e.g. for \(i = 1\), \((d_1, d_2) \neq (1, 0), (1, -1 - βγ)\)).

Introduced

2016, MR3490085. The cocycle twist of Davies' \(\mathrm{S}_{d,i}\) (equation 6.1.8 / 6.1.4 twisted) by the Klein four-group; a finite member of the family of which the \(\mathrm{S}_\infty\) twist 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 >;