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deformed skew \((x_2x_1;\ x_3x_4)\)

A quantum (skew) polynomial ring with a single quadratic term added: the \(x_2 x_1\) relation acquires the cross-term \(-t\,x_3 x_4\).

Relations

• \(x_2 x_1 - q_{12} x_1 x_2 - t x_3 x_4\)
• \(x_3 x_1 - q_{13} x_1 x_3\)
• \(x_4 x_1 - q_{14} x_1 x_4\)
• \(x_3 x_2 - q_{23} x_2 x_3\)
• \(x_4 x_2 - (1/(q_{13} q_{23} q_{14})) x_2 x_4\)
• \(x_4 x_3 - (1/(q_{13} q_{23})) x_3 x_4\)

Parameters

• \(q_{12}, q_{13}, q_{14}, q_{23}\) — skew-commutation parameters.
• \(t\) — coefficient of the deforming term.

Point scheme

a union of 5 lines (dimension \(1\)).

Line scheme

two planes, two conics 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 \(A\)

\(\mathrm{HH}^\bullet_0(A) = (1, 3, 4, 3)\)

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

\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 2, 4, 7)\)

Kodaira–Spencer

rank \(4\)
injective: no
surjective: yes

Nakayama automorphism

\(\nu = \operatorname{diag}\!\left(\tfrac{1}{q_{12}q_{13}q_{14}},q_{12}q_{13}q_{14},q_{13}^{2}q_{23}^{2},\tfrac{1}{q_{13}^{2}q_{23}^{2}}\right)\).
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[q12, q13, q14, q23, t]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
  x2*x1 - q12*x1*x2 - t*x3*x4,
  x3*x1 - q13*x1*x3,
  x4*x1 - q14*x1*x4,
  x3*x2 - q23*x2*x3,
  x4*x2 - (1/(q13*q23*q14))*x2*x4,
  x4*x3 - (1/(q13*q23))*x3*x4
};
B = A/I;

PolyRing := FunctionField(Rationals, ["q12", "q13", "q14", "q23", "t"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
q12 := indets[1];;
q13 := indets[2];;
q14 := indets[3];;
q23 := 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 - q12*x1*x2 - t*x3*x4,
  x3*x1 - q13*x1*x3,
  x4*x1 - q14*x1*x4,
  x3*x2 - q23*x2*x3,
  x4*x2 - (1/(q13*q23*q14))*x2*x4,
  x4*x3 - (1/(q13*q23))*x3*x4
];;
A := kQ / rels;;

// untested (Magma not available here)
K<q12, q13, q14, q23, t> := RationalFunctionField(Rationals(), 5);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
  x2*x1 - q12*x1*x2 - t*x3*x4,
  x3*x1 - q13*x1*x3,
  x4*x1 - q14*x1*x4,
  x3*x2 - q23*x2*x3,
  x4*x2 - (1/(q13*q23*q14))*x2*x4,
  x4*x3 - (1/(q13*q23))*x3*x4
];
A := quo< F | rels >;