$\mathrm{S}_\infty$ twist

A cocycle twist of Stafford's $\mathrm{S}_\infty$ by the Klein four-group. The surviving Sklyanin relations are twisted (commutators exchanged with anticommutators) and the signs in the two quadrics change.

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 counting multiplicity for the general member in characteristic zero (Davies, Proposition 6.1.19) (dimension $0$).

Line scheme

five quartic curves (dimension $1$).

Centre

$\dim \mathrm{Z}_2(A)=1$ for the general member in characteristic zero, spanned by $x_1x_4+x_4x_1+x_2x_3+x_3x_2$ (computed over $\mathbb{Q}(\beta,\gamma)$) .

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

2014, arXiv:1502.06101. Davies's 2014 PhD thesis, §6.1, Lemma 6.1.6 and equation (6.1.7).

References

Davies, A. (2015). Cocycle Twists of Algebras.

arXiv:1502.06101

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 >;

4d-AS-regular

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

References

Code