4d-AS-regular

the classification of 4-dimensional quadratic Artin–Schelter regular algebras

← all families

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

As for \(\mathrm{S}_\infty\), 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 \(0\)).
Line scheme
five quartic curves (dimension \(1\)).
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 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 \(2\)
injective: yes
surjective: no
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. The cocycle twist of \(\mathrm{S}_\infty\) by the Klein four-group, in the same family of exotic elliptic algebras studied by Davies and by Chirvasitu–Smith.

References

Davies, A. (2016). Cocycle twists of 4-dimensional Sklyanin algebras. J. Algebra, 457, 323–360.
MR3490085 doi
Chirvasitu, A., & Smith, S. P. (2019). Exotic elliptic algebras. Trans. Amer. Math. Soc., 371(1), 279–333.
MR3885145 doi

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 >;