deformed skew \((x_2x_1, x_3x_2;\ x_3^2, x_4^2)\)

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

Relations

\(x_2 x_1 - q_{12} x_1 x_2 - t x_3^2\)
\(x_3 x_1 - (1/(q_{24}^6 q_{12})) x_1 x_3\)
\(x_4 x_1 - (1/q_{24}^3) x_1 x_4\)
\(x_3 x_2 - (q_{24}^6 q_{12}) x_2 x_3 - s x_4^2\)
\(x_4 x_2 - q_{24} x_2 x_4\)
\(x_4 x_3 - (1/q_{24}) x_3 x_4\)

Parameters

\(q_{12}, q_{24}\) — skew-commutation parameters.
\(t, s\) — coefficients of the two deforming terms.

Point scheme

a line and a conic (dimension \(1\)).

Normal elements

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

Hochschild cohomology

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

Kodaira–Spencer

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

Introduced

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

References

arXiv:2511.08390

Code

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

needsPackage "AssociativeAlgebras"
K = frac(QQ[q12, q24, s, t]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
  x2*x1 - q12*x1*x2 - t*x3^2,
  x3*x1 - (1/*(q24^6*q12))*x1*x3,
  x4*x1 - (1/q24^3)*x1*x4,
  x3*x2 - (q24^6*q12)*x2*x3 - s*x4^2,
  x4*x2 - q24*x2*x4,
  x4*x3 - (1/q24)*x3*x4
};
B = A/I;

PolyRing := FunctionField(Rationals, ["q12", "q24", "s", "t"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
q12 := indets[1];;
q24 := 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 - t*x3^2,
  x3*x1 - (1/*(q24^6*q12))*x1*x3,
  x4*x1 - (1/q24^3)*x1*x4,
  x3*x2 - (q24^6*q12)*x2*x3 - s*x4^2,
  x4*x2 - q24*x2*x4,
  x4*x3 - (1/q24)*x3*x4
];;
A := kQ / rels;;

// untested (Magma not available here)
K<q12, q24, s, t> := RationalFunctionField(Rationals(), 4);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
  x2*x1 - q12*x1*x2 - t*x3^2,
  x3*x1 - (1/*(q24^6*q12))*x1*x3,
  x4*x1 - (1/q24^3)*x1*x4,
  x3*x2 - (q24^6*q12)*x2*x3 - s*x4^2,
  x4*x2 - q24*x2*x4,
  x4*x3 - (1/q24)*x3*x4
];
A := quo< F | rels >;

4d-AS-regular

deformed skew \((x_2x_1, x_3x_2;\ x_3^2, x_4^2)\)

Code