Goetz–Kirkman–Moore–Vashaw S

A quadratic algebra with no scalar parameters coming from a dual reflection group: four of the relations identify a pair of mixed monomials, while two relations equate a product with a square \(x_3^2\) or \(x_4^2\). The Kodaira–Spencer map vanishes, so the family is a rigid point of the moduli space.

Relations

\(x_1 x_2 - x_3 x_3\)
\(x_1 x_3 - x_2 x_4\)
\(x_1 x_4 - x_4 x_2\)
\(x_2 x_3 - x_3 x_1\)
\(x_3 x_2 - x_4 x_1\)
\(x_2 x_1 - x_4 x_4\)

Centre

\(\dim \operatorname{Z}(A)_2 = 1\), \(\dim \operatorname{Z}(A)_3 = 0\), \(\dim \operatorname{Z}(A)_4 = 4\).

Hochschild cohomology of \(A\)

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

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

\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 0, 15, 20)\)

Kodaira–Spencer

rank \(0\)
injective: yes
surjective: no

Introduced

2024, arXiv:2410.08959. The algebra "S" of Goetz–Kirkman–Moore–Vashaw, one of three dimension-four Artin–Schelter regular algebras they construct from a dual reflection group of order 16. It is a rigid (parameter-free) example.

References

Goetz, P., Kirkman, E. E., Moore, W. F., & Vashaw, K. B. (2024). Some Artin–Schelter Regular Algebras From Dual Reflection Groups and their Geometry.

arXiv:2410.08959

Code

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

needsPackage "AssociativeAlgebras"
K = QQ;
A = K<|x1,x2,x3,x4|>;
I = ideal {
  x1*x2 - x3*x3,
  x1*x3 - x2*x4,
  x1*x4 - x4*x2,
  x2*x3 - x3*x1,
  x3*x2 - x4*x1,
  x2*x1 - x4*x4
};
B = A/I;

PolyRing := Rationals;;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
  x1*x2 - x3*x3,
  x1*x3 - x2*x4,
  x1*x4 - x4*x2,
  x2*x3 - x3*x1,
  x3*x2 - x4*x1,
  x2*x1 - x4*x4
];;
A := kQ / rels;;

// untested (Magma not available here)
K := Rationals();
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
  x1*x2 - x3*x3,
  x1*x3 - x2*x4,
  x1*x4 - x4*x2,
  x2*x3 - x3*x1,
  x3*x2 - x4*x1,
  x2*x1 - x4*x4
];
A := quo< F | rels >;

4d-AS-regular

Goetz–Kirkman–Moore–Vashaw S

References

Code