central extension of Sklyanin twist

Obtained by applying a 2-cocycle (Klein four-group) twist to a central extension of a 3-dimensional regular algebra. The variable \(x_3\) plays the role of the adjoined central-type generator: it commutes with each of \(x_1, x_2, x_4\), while the first three relations \(q\)-commute the remaining pairs and absorb quadratic corrections \(b x_\bullet^2 + c_\bullet x_3^2\).

Relations

\(x_2 x_1 + q x_1 x_2 + b x_4^2 + c_0 x_3^2\)
\(x_4 x_2 + q x_2 x_4 + b x_1^2 + c_1 x_3^2\)
\(x_1 x_4 + q x_4 x_1 + b x_2^2 + c_2 x_3^2\)
\(x_4 x_3 + x_3 x_4\)
\(x_1 x_3 + x_3 x_1\)
\(x_2 x_3 + x_3 x_2\)

Parameters

\(q, b, c_0, c_1, c_2\) — Five scalar parameters of the twisted central extension.

Centre

\(\dim \operatorname{Z}(A)_2 = 1\), \(\dim \operatorname{Z}(A)_3 = 0\), \(\dim \operatorname{Z}(A)_4 = 1\).

Hochschild cohomology of \(A\)

\(\mathrm{HH}^\bullet_0(A) = (1, 1, 4, 9)\)

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

\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 0, 7, 12)\)

Kodaira–Spencer

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

Introduced

2025, MR1429334. A cocycle twist of the central extension of a 3-dimensional Sklyanin-type Artin–Schelter regular algebra (Le Bruyn–Smith–Van den Bergh). It appears as "Central Extensions twist" in Tables 7 and 8 of arXiv:2511.08390, attributed there to Remark 5.13, and is the cocycle twist of the central-extension family in the same spirit that the Sklyanin twist relates to the Sklyanin algebra.

References

Le Bruyn, L., Smith, S. P., & Van den Bergh, M. (1996). Central extensions of three-dimensional Artin-Schelter regular algebras. Math. Z., 222(2), 171–212.

MR1429334 doi

Code

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

needsPackage "AssociativeAlgebras"
K = frac(QQ[b, c0, c1, c2, q]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
  x2*x1 + q*x1*x2 + b*x4^2 + c0*x3^2,
  x4*x2 + q*x2*x4 + b*x1^2 + c1*x3^2,
  x1*x4 + q*x4*x1 + b*x2^2 + c2*x3^2,
  x4*x3 + x3*x4,
  x1*x3 + x3*x1,
  x2*x3 + x3*x2
};
B = A/I;

PolyRing := FunctionField(Rationals, ["b", "c0", "c1", "c2", "q"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
b := indets[1];;
c0 := indets[2];;
c1 := indets[3];;
c2 := indets[4];;
q := indets[5];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
  x2*x1 + q*x1*x2 + b*x4^2 + c0*x3^2,
  x4*x2 + q*x2*x4 + b*x1^2 + c1*x3^2,
  x1*x4 + q*x4*x1 + b*x2^2 + c2*x3^2,
  x4*x3 + x3*x4,
  x1*x3 + x3*x1,
  x2*x3 + x3*x2
];;
A := kQ / rels;;

// untested (Magma not available here)
K<b, c0, c1, c2, q> := RationalFunctionField(Rationals(), 5);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
  x2*x1 + q*x1*x2 + b*x4^2 + c0*x3^2,
  x4*x2 + q*x2*x4 + b*x1^2 + c1*x3^2,
  x1*x4 + q*x4*x1 + b*x2^2 + c2*x3^2,
  x4*x3 + x3*x4,
  x1*x3 + x3*x1,
  x2*x3 + x3*x2
];
A := quo< F | rels >;

4d-AS-regular

central extension of Sklyanin twist

References

Code