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deformed skew \((x_3x_2;\ x_4^2, x_1^2)\)

A quantum (skew) polynomial ring with two quadratic terms added to one relation: \(x_3 x_2\) acquires \(-t\,x_4^2 - s\,x_1^2\).

Relations

• \(x_2 x_1 - (1/q_{13}) x_1 x_2\)
• \(x_3 x_1 - q_{13} x_1 x_3\)
• \(x_4 x_1 - x_1 x_4\)
• \(x_3 x_2 - q_{23} x_2 x_3 - t x_4^2 - s x_1^2\)
• \(x_4 x_2 - (1/q_{34}) x_2 x_4\)
• \(x_4 x_3 - q_{34} x_3 x_4\)

Parameters

• \(q_{13}, q_{23}, q_{34}\) — skew-commutation parameters.
• \(t, s\) — coefficients of the two deforming terms.

Point scheme

a line, two conics and two points (dimension \(1\)).

Line scheme

two planes, a quartic curve and two 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 \(A\)

\(\mathrm{HH}^\bullet_0(A) = (1, 2, 4, 8)\)

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

\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 1, 3, 7)\)

Kodaira–Spencer

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

Nakayama automorphism

\(\nu = \operatorname{diag}\!\left(1,\tfrac{q_{34}}{q_{13}q_{23}},\tfrac{q_{13}q_{23}}{q_{34}},1\right)\).
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

Code

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

• Macaulay2
• GAP
• Magma

needsPackage "AssociativeAlgebras"
K = frac(QQ[q13, q23, q34, s, t]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
  x2*x1 - (1/q13)*x1*x2,
  x3*x1 - q13*x1*x3,
  x4*x1 - x1*x4,
  x3*x2 - q23*x2*x3 - t*x4^2 - s*x1^2,
  x4*x2 - (1/q34)*x2*x4,
  x4*x3 - q34*x3*x4
};
B = A/I;

PolyRing := FunctionField(Rationals, ["q13", "q23", "q34", "s", "t"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
q13 := indets[1];;
q23 := indets[2];;
q34 := indets[3];;
s := indets[4];;
t := indets[5];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
  x2*x1 - (1/q13)*x1*x2,
  x3*x1 - q13*x1*x3,
  x4*x1 - x1*x4,
  x3*x2 - q23*x2*x3 - t*x4^2 - s*x1^2,
  x4*x2 - (1/q34)*x2*x4,
  x4*x3 - q34*x3*x4
];;
A := kQ / rels;;

// untested (Magma not available here)
K<q13, q23, q34, s, t> := RationalFunctionField(Rationals(), 5);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
  x2*x1 - (1/q13)*x1*x2,
  x3*x1 - q13*x1*x3,
  x4*x1 - x1*x4,
  x3*x2 - q23*x2*x3 - t*x4^2 - s*x1^2,
  x4*x2 - (1/q34)*x2*x4,
  x4*x3 - q34*x3*x4
];
A := quo< F | rels >;