4d-AS-regular

the classification of 4-dimensional Artin–Schelter regular algebras

← all families

Generalized Clifford 1

A graded skew Clifford algebra: the symmetric Clifford relations are twisted by roots of unity, so \(x_4, x_1\) and \(x_3, x_2\) skew-commute while the remaining relations carry quadratic corrections. Such algebras are regular when the defining quadrics are normalizing and base-point free.

Relations
  • \(x_4 x_1 - α x_1 x_4\)
  • \(x_3 x_2 - β x_2 x_3\)
  • \(x_3^2 - x_1^2\)
  • \(x_4^2 - x_2^2\)
  • \(x_3 x_1 - x_1 x_3 + x_2^2\)
  • \(x_4 x_2 - x_2 x_4 + γ^2 x_1^2\)
Parameters
  • \(α, β\) — Primitive 4th roots of unity (\(α^2 = β^2 = -1\)).
  • \(γ\) — A nonzero scalar.
Point scheme
20 points (dimension \(0\)).
Centre
\(\dim \operatorname{Z}(A)_2 = 0\), \(\dim \operatorname{Z}(A)_3 = 0\), \(\dim \operatorname{Z}(A)_4 = 2\).
Normal elements
normal locus dimension \(-1\) (degree 1), \(0\) (degree 2).
Hochschild cohomology of \(A\)
\(\mathrm{HH}^\bullet_0(A) = (1, 1, 1, 7)\)
Hochschild cohomology of \(\operatorname{qgr} A\)
\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 0, 16, 21)\)
Kodaira–Spencer
rank \(1\)
injective: yes
surjective: yes
Introduced
2010, MR2580455. Example 1 of Cassidy–Vancliff. Of the three, it is a genuinely *skew* graded Clifford algebra: the generators \(q\)-commute by primitive 4th roots of unity. Its centre is trivial in degree 2 (central elements first appear in degree 4).

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