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\(\mathrm{A}_5\)

A four-parameter quadratic algebra defined over a field containing \(\sqrt{-1}\). Three relations make \(x_4\) skew/normalize against \(x_1, x_2, x_3\) (with the coefficient \(i\) appearing in the \(x_3, x_4\) relation), and two further relations impose quadratic identities among the squares \(x_1^2, x_2^2, x_3^2\).

Relations

• \(d x_1 x_4 + x_4 x_1\)
• \(d x_2 x_4 - x_4 x_2\)
• \(a_1 x_1^2 + a_4 d^2 x_2^2\)
• \(d x_3 x_4 - i x_4 x_3\)
• \(a_4 x_1 x_2 + a_4 x_2 x_1 - a_7 x_3^2\)
• \(x_2 x_3 + x_3 x_2\)

Parameters

• \(d, a_1, a_4, a_7\) — Four scalars in the base field. The relations also involve the primitive fourth root of unity \(i\) (with \(i^2 = -1\)).

Centre

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

Hochschild cohomology of \(A\)

\(\mathrm{HH}^\bullet_0(A) = (1, 2, 1, 0)\)

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

\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 1, 1, 5)\)

Kodaira–Spencer

rank \(1\)
injective: no
surjective: yes

Notes

\(\mathrm{A}_5\) is defined over a field containing \(\sqrt{-1}\) (here \(\mathbb{Q}(i)\), with \(i\) the primitive fourth root of unity appearing in the relations); its invariants are computed over that field.

Introduced

2025, arXiv:2511.08390. \(\mathrm{A}_5\) appears among the families mapping densely onto irreducible components of the moduli space of regular algebras (its Kodaira–Spencer map is surjective of rank 1). No earlier bibliographic source is identified for it.

References

Code

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

• Macaulay2
• GAP
• Magma

needsPackage "AssociativeAlgebras"
K = frac(QQ[a1, a4, a7, d, ii]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
  d*x1*x4 + x4*x1,
  d*x2*x4 - x4*x2,
  a1*x1^2 + a4*d^2*x2^2,
  d*x3*x4 - ii*x4*x3,
  a4*x1*x2 + a4*x2*x1 - a7*x3^2,
  x2*x3 + x3*x2
};
B = A/I;

PolyRing := FunctionField(Rationals, ["a1", "a4", "a7", "d", "ii"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
a1 := indets[1];;
a4 := indets[2];;
a7 := indets[3];;
d := indets[4];;
ii := indets[5];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
  d*x1*x4 + x4*x1,
  d*x2*x4 - x4*x2,
  a1*x1^2 + a4*d^2*x2^2,
  d*x3*x4 - ii*x4*x3,
  a4*x1*x2 + a4*x2*x1 - a7*x3^2,
  x2*x3 + x3*x2
];;
A := kQ / rels;;

// untested (Magma not available here)
K<a1, a4, a7, d, ii> := RationalFunctionField(Rationals(), 5);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
  d*x1*x4 + x4*x1,
  d*x2*x4 - x4*x2,
  a1*x1^2 + a4*d^2*x2^2,
  d*x3*x4 - ii*x4*x3,
  a4*x1*x2 + a4*x2*x1 - a7*x3^2,
  x2*x3 + x3*x2
];
A := quo< F | rels >;