deformed skew \((x_3x_1, x_3x_2;\ x_4^2)\)

A quantum (skew) polynomial ring deformed in two relations by \(x_4^2\): the \(x_3 x_1\) relation acquires \(-t\,x_4^2\) and the \(x_3 x_2\) relation \(-s\,x_4^2\).

Relations

\(x_2 x_1 - q_{12} x_1 x_2\)
\(x_3 x_1 - (q_{14}^2/q_{12}) x_1 x_3 - t x_4^2\)
\(x_4 x_1 - q_{14} x_1 x_4\)
\(x_3 x_2 - (q_{14}^2 q_{12}) x_2 x_3 - s x_4^2\)
\(x_4 x_2 - q_{14} x_2 x_4\)
\(x_4 x_3 - (1/q_{14}) x_3 x_4\)

Parameters

\(q_{12}, q_{14}\) — skew-commutation parameters.
\(t, s\) — coefficients of the two deforming terms.

Point scheme

a union of 3 lines (dimension \(1\)).

Line scheme

a plane, two conics and three lines (dimension \(2\)).

Centre

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

Normal elements

normal locus dimension \(0\) (degree 1), \(0\) (degree 2).

Hochschild cohomology of \(A\)

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

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

\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 1, 8, 12)\)

Kodaira–Spencer

rank \(2\)
injective: no
surjective: yes

Nakayama automorphism

\(\nu = \operatorname{diag}(q_{14}^{-3},q_{14}^{-3},q_{14}^{5},q_{14})\).
Depends on the parameters.
Homological determinant \(\det \nu = 1\).
Computed exactly over \(\mathbb{Q}(\text{params})\).

Introduced

arXiv:2511.08390. One of the deformed skew polynomial (\(\mathcal{F}\)) families; its earlier origin is not pinned down here.

References

Bhatoy, V., Ingalls, C., LaRoche, F., & Nookala, R. (2025). Some components of the moduli space of Koszul Artin–Schelter regular algebras of dimension four.

arXiv:2511.08390

Code

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

needsPackage "AssociativeAlgebras"
K = frac(QQ[q12, q14, s, t]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
  x2*x1 - q12*x1*x2,
  x3*x1 - (q14^2/q12)*x1*x3 - t*x4^2,
  x4*x1 - q14*x1*x4,
  x3*x2 - (q14^2*q12)*x2*x3 - s*x4^2,
  x4*x2 - q14*x2*x4,
  x4*x3 - (1/q14)*x3*x4
};
B = A/I;

PolyRing := FunctionField(Rationals, ["q12", "q14", "s", "t"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
q12 := indets[1];;
q14 := indets[2];;
s := indets[3];;
t := indets[4];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
  x2*x1 - q12*x1*x2,
  x3*x1 - (q14^2/q12)*x1*x3 - t*x4^2,
  x4*x1 - q14*x1*x4,
  x3*x2 - (q14^2*q12)*x2*x3 - s*x4^2,
  x4*x2 - q14*x2*x4,
  x4*x3 - (1/q14)*x3*x4
];;
A := kQ / rels;;

// untested (Magma not available here)
K<q12, q14, s, t> := RationalFunctionField(Rationals(), 4);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
  x2*x1 - q12*x1*x2,
  x3*x1 - (q14^2/q12)*x1*x3 - t*x4^2,
  x4*x1 - q14*x1*x4,
  x3*x2 - (q14^2*q12)*x2*x3 - s*x4^2,
  x4*x2 - q14*x2*x4,
  x4*x3 - (1/q14)*x3*x4
];
A := quo< F | rels >;

4d-AS-regular

deformed skew \((x_3x_1, x_3x_2;\ x_4^2)\)

References

Code