4d-AS-regular

the classification of 4-dimensional Artin–Schelter regular algebras

← all families

Clifford

Each pair of generators anticommutes up to a quadratic correction that is a linear combination of the squares \(x_ℓ^2\). Equivalently the algebra is determined by a symmetric \(4 \times 4\) matrix of quadratic forms; it is Artin–Schelter regular exactly when the associated system of quadrics is base-point free. The four squares span a \(4\)-dimensional space of central quadrics.

Relations
  • \(x_1 x_2 + x_2 x_1 + a_{121} x_1^2 + a_{122} x_2^2 + a_{123} x_3^2 + a_{124} x_4^2\)
  • \(x_1 x_3 + x_3 x_1 + a_{131} x_1^2 + a_{132} x_2^2 + a_{133} x_3^2 + a_{134} x_4^2\)
  • \(x_1 x_4 + x_4 x_1 + a_{141} x_1^2 + a_{142} x_2^2 + a_{143} x_3^2 + a_{144} x_4^2\)
  • \(x_2 x_3 + x_3 x_2 + a_{231} x_1^2 + a_{232} x_2^2 + a_{233} x_3^2 + a_{234} x_4^2\)
  • \(x_2 x_4 + x_4 x_2 + a_{241} x_1^2 + a_{242} x_2^2 + a_{243} x_3^2 + a_{244} x_4^2\)
  • \(x_3 x_4 + x_4 x_3 + a_{341} x_1^2 + a_{342} x_2^2 + a_{343} x_3^2 + a_{344} x_4^2\)
Parameters
  • \(a_{ijℓ}\) — For each pair \(i < j\), the four coefficients of \(x_1^2, x_2^2, x_3^2, x_4^2\) in the corresponding relation — equivalently four symmetric matrices of quadrics (24 scalars).
Point scheme
20 points (dimension \(0\)).
Centre
\(\dim \operatorname{Z}(A)_2 = 4\) (the four squares are central).
Normal elements
normal locus dimension \(-1\) (degree 1), \(3\) (degree 2).
Hochschild cohomology
\(\mathrm{HH}^\bullet_0(A)\) \(= (1, 1, 9, 19)\)
\(\mathrm{HH}^\bullet(\operatorname{qgr} A)\) \(= (1, 0, 15, 20)\)
Kodaira–Spencer
rank \(9\); injective: no; surjective: yes.
Introduced
1995, MR1356364 . A graded Clifford algebra; the graded skew generalisation is due to Cassidy–Vancliff.
References
MR1356364 , MR2580455 , arXiv:2511.08390

Code

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

needsPackage "AssociativeAlgebras"
K = frac(QQ[a121, a122, a123, a124, a131, a132, a133, a134, a141, a142, a143, a144, a231, a232, a233, a234, a241, a242, a243, a244, a341, a342, a343, a344]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
  x1*x2 + x2*x1 + a121*x1^2 + a122*x2^2 + a123*x3^2 + a124*x4^2,
  x1*x3 + x3*x1 + a131*x1^2 + a132*x2^2 + a133*x3^2 + a134*x4^2,
  x1*x4 + x4*x1 + a141*x1^2 + a142*x2^2 + a143*x3^2 + a144*x4^2,
  x2*x3 + x3*x2 + a231*x1^2 + a232*x2^2 + a233*x3^2 + a234*x4^2,
  x2*x4 + x4*x2 + a241*x1^2 + a242*x2^2 + a243*x3^2 + a244*x4^2,
  x3*x4 + x4*x3 + a341*x1^2 + a342*x2^2 + a343*x3^2 + a344*x4^2
};
B = A/I;
PolyRing := FunctionField(Rationals, ["a121", "a122", "a123", "a124", "a131", "a132", "a133", "a134", "a141", "a142", "a143", "a144", "a231", "a232", "a233", "a234", "a241", "a242", "a243", "a244", "a341", "a342", "a343", "a344"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
a121 := indets[1];;
a122 := indets[2];;
a123 := indets[3];;
a124 := indets[4];;
a131 := indets[5];;
a132 := indets[6];;
a133 := indets[7];;
a134 := indets[8];;
a141 := indets[9];;
a142 := indets[10];;
a143 := indets[11];;
a144 := indets[12];;
a231 := indets[13];;
a232 := indets[14];;
a233 := indets[15];;
a234 := indets[16];;
a241 := indets[17];;
a242 := indets[18];;
a243 := indets[19];;
a244 := indets[20];;
a341 := indets[21];;
a342 := indets[22];;
a343 := indets[23];;
a344 := indets[24];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
  x1*x2 + x2*x1 + a121*x1^2 + a122*x2^2 + a123*x3^2 + a124*x4^2,
  x1*x3 + x3*x1 + a131*x1^2 + a132*x2^2 + a133*x3^2 + a134*x4^2,
  x1*x4 + x4*x1 + a141*x1^2 + a142*x2^2 + a143*x3^2 + a144*x4^2,
  x2*x3 + x3*x2 + a231*x1^2 + a232*x2^2 + a233*x3^2 + a234*x4^2,
  x2*x4 + x4*x2 + a241*x1^2 + a242*x2^2 + a243*x3^2 + a244*x4^2,
  x3*x4 + x4*x3 + a341*x1^2 + a342*x2^2 + a343*x3^2 + a344*x4^2
];;
A := kQ / rels;;
// untested (Magma not available here)
K<a121, a122, a123, a124, a131, a132, a133, a134, a141, a142, a143, a144, a231, a232, a233, a234, a241, a242, a243, a244, a341, a342, a343, a344> := RationalFunctionField(Rationals(), 24);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
  x1*x2 + x2*x1 + a121*x1^2 + a122*x2^2 + a123*x3^2 + a124*x4^2,
  x1*x3 + x3*x1 + a131*x1^2 + a132*x2^2 + a133*x3^2 + a134*x4^2,
  x1*x4 + x4*x1 + a141*x1^2 + a142*x2^2 + a143*x3^2 + a144*x4^2,
  x2*x3 + x3*x2 + a231*x1^2 + a232*x2^2 + a233*x3^2 + a234*x4^2,
  x2*x4 + x4*x2 + a241*x1^2 + a242*x2^2 + a243*x3^2 + a244*x4^2,
  x3*x4 + x4*x3 + a341*x1^2 + a342*x2^2 + a343*x3^2 + a344*x4^2
];
A := quo< F | rels >;