4d-AS-regular

the classification of 4-dimensional Artin–Schelter regular algebras

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Generalized Clifford 2

Another graded skew Clifford algebra, presented through symmetric and skew-symmetric relations whose quadratic corrections are multiples of the squares \(x_2^2, x_3^2\). It carries a \(2\)-dimensional space of central quadrics.

Relations
  • \(x_3 x_1 + x_1 x_3 - β_2 x_2^2\)
  • \(x_4 x_1 + x_1 x_4 - α_2 x_3^2\)
  • \(x_2 x_3 - x_3 x_2\)
  • \(x_4^2 - x_2^2\)
  • \(x_4 x_2 + x_2 x_4 - x_3^2\)
  • \(α_1 x_3^2 + β_1 x_2^2 - x_1^2\)
Parameters
  • \(α_1, α_2, β_1, β_2\) — Four scalars.
Point scheme
20 points (dimension \(0\)).
Centre
\(\dim \operatorname{Z}(A)_2 = 2\).
Normal elements
normal locus dimension \(-1\) (degree 1), \(1\) (degree 2).
Hochschild cohomology of \(A\)
\(\mathrm{HH}^\bullet_0(A) = (1, 1, 9, 19)\)
Hochschild cohomology of \(\operatorname{qgr} A\)
\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 0, 16, 21)\)
Kodaira–Spencer
rank \(3\)
injective: no
surjective: no
Introduced
2010, MR2580455. Example 2 of Cassidy–Vancliff. It is the example that is an ordinary (non-skew) graded Clifford algebra and a complete intersection, and so has by far the largest centre of the three — a \(2\)-dimensional space of central quadrics, with centre Hilbert series \(1 + 2t^2 + 5t^4 + 8t^6 + \cdots\).

References

Cassidy, T., & Vancliff, M. (2010). Generalizations of graded Clifford algebras and of complete intersections. J. Lond. Math. Soc. (2), 81(1), 91–112.
MR2580455 doi

Code

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

needsPackage "AssociativeAlgebras"
K = frac(QQ[alpha1, alpha2, beta1, beta2]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
  x3*x1 + x1*x3 - beta2*x2^2,
  x4*x1 + x1*x4 - alpha2*x3^2,
  x2*x3 - x3*x2,
  x4^2 - x2^2,
  x4*x2 + x2*x4 - x3^2,
  alpha1*x3^2 + beta1*x2^2 - x1^2
};
B = A/I;
PolyRing := FunctionField(Rationals, ["alpha1", "alpha2", "beta1", "beta2"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
alpha1 := indets[1];;
alpha2 := indets[2];;
beta1 := indets[3];;
beta2 := indets[4];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
  x3*x1 + x1*x3 - beta2*x2^2,
  x4*x1 + x1*x4 - alpha2*x3^2,
  x2*x3 - x3*x2,
  x4^2 - x2^2,
  x4*x2 + x2*x4 - x3^2,
  alpha1*x3^2 + beta1*x2^2 - x1^2
];;
A := kQ / rels;;
// untested (Magma not available here)
K<alpha1, alpha2, beta1, beta2> := RationalFunctionField(Rationals(), 4);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
  x3*x1 + x1*x3 - beta2*x2^2,
  x4*x1 + x1*x4 - alpha2*x3^2,
  x2*x3 - x3*x2,
  x4^2 - x2^2,
  x4*x2 + x2*x4 - x3^2,
  alpha1*x3^2 + beta1*x2^2 - x1^2
];
A := quo< F | rels >;