4d-AS-regular

the classification of 4-dimensional Artin–Schelter regular algebras

← all families

Generalized Clifford 3

A graded skew Clifford algebra in which three pairs \(q\)-commute and the remaining relations couple the squares \(x_1^2, x_4^2\) to the products \(x_2 x_3, x_2 x_4\). Regularity holds only on a constraint subvariety of the parameters.

Relations
  • \(x_1 x_3 + u_{13} x_3 x_1\)
  • \(x_1 x_4 + u_{14} x_4 x_1\)
  • \(x_3 x_4 + u_{34} x_4 x_3\)
  • \(x_4^2 - x_2^2\)
  • \(x_2 x_3 + x_3 x_2 + x_4^2\)
  • \(x_2 x_4 + u_{24} x_4 x_2 + x_1^2\)
Parameters
  • \(u_{13}, u_{14}, u_{24}, u_{34}\) — Four scalars subject to a constraint variety for regularity (\(u_{34}^2 = 1\), \(u_{34} = u_{24}\), \(u_{14}^2 = u_{13}^2\)).
Point scheme
20 points (dimension \(0\)).
Hochschild cohomology of \(A\)
\(\mathrm{HH}^\bullet_0(A) = (1, 1, 9, 7)\)
Kodaira–Spencer
rank \(4\)
injective: yes
surjective: no
Notes
Artin–Schelter regular only at special (root-of-unity) values of the parameters; the invariants above are for such a point and depend on the choice (in particular the centre is not generic), so they should be read with care.
Introduced
2010, MR2580455. Example 3 of Cassidy–Vancliff. Like Example 1 it is a *skew* graded Clifford algebra, but it is Artin–Schelter regular only on a constraint subvariety of its parameter space; off that variety it fails to be regular.

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[u13, u14, u24, u34]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
  x1*x3 + u13*x3*x1,
  x1*x4 + u14*x4*x1,
  x3*x4 + u34*x4*x3,
  x4^2 - x2^2,
  x2*x3 + x3*x2 + x4^2,
  x2*x4 + u24*x4*x2 + x1^2
};
B = A/I;
PolyRing := FunctionField(Rationals, ["u13", "u14", "u24", "u34"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
u13 := indets[1];;
u14 := indets[2];;
u24 := indets[3];;
u34 := indets[4];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
  x1*x3 + u13*x3*x1,
  x1*x4 + u14*x4*x1,
  x3*x4 + u34*x4*x3,
  x4^2 - x2^2,
  x2*x3 + x3*x2 + x4^2,
  x2*x4 + u24*x4*x2 + x1^2
];;
A := kQ / rels;;
// untested (Magma not available here)
K<u13, u14, u24, u34> := RationalFunctionField(Rationals(), 4);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
  x1*x3 + u13*x3*x1,
  x1*x4 + u14*x4*x1,
  x3*x4 + u34*x4*x3,
  x4^2 - x2^2,
  x2*x3 + x3*x2 + x4^2,
  x2*x4 + u24*x4*x2 + x1^2
];
A := quo< F | rels >;