skew

The skew polynomial ring: each pair of generators \(q\)-commutes, \(x_i x_j = q_{ij}\, x_j x_i\). This is the most "abelian" noncommutative \(\mathbb{P}^3\), a toric deformation of the polynomial ring. Its point scheme is the \(1\)-skeleton of the coordinate tetrahedron — the six edges (a copy of \(K_4\)), two edges meeting iff they share a vertex.

Relations

\(x_1 x_2 - q_{12} x_2 x_1\)
\(x_1 x_3 - q_{13} x_3 x_1\)
\(x_1 x_4 - q_{14} x_4 x_1\)
\(x_2 x_3 - q_{23} x_3 x_2\)
\(x_2 x_4 - q_{24} x_4 x_2\)
\(x_3 x_4 - q_{34} x_4 x_3\)

Parameters

\(q_{ij}\) — one nonzero scalar for each pair \(i < j\) (six in total).

Point scheme

six lines, the tetrahedron skeleton \(K_4\) (dimension \(1\)).

Line scheme

four planes (dimension \(2\)).

Centre

trivial in low degrees.

Normal elements

normal locus dimension \(0\) (degree 1), \(0\) (degree 2).

Hochschild cohomology of \(A\)

\(\mathrm{HH}^\bullet_0(A) = (1, 4, 6, 4)\)

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

\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 3, 3, 5)\)

Kodaira–Spencer

rank \(6\)
injective: yes
surjective: yes

Nakayama automorphism

\(\nu = \operatorname{diag}\!\left(q_{12}q_{13}q_{14},\tfrac{q_{23}q_{24}}{q_{12}},\tfrac{q_{34}}{q_{13}q_{23}},\tfrac{1}{q_{14}q_{24}q_{34}}\right)\).
Depends on the parameters.
Homological determinant \(\det \nu = 1\).
Computed exactly over \(\mathbb{Q}(\text{params})\).

Introduced

1990, MR3527537. The generic quantum (skew) polynomial ring; its point variety was determined by Belmans–De Laet–Le Bruyn. It also occurs as the four-generator graded skew Clifford algebra.

References

Belmans, P., De Laet, K., & Le Bruyn, L. (2016). The point variety of quantum polynomial rings. J. Algebra, 463, 10–22.

MR3527537 doi

Artin, M., Tate, J., & Van den Bergh, M. (1990). Some algebras associated to automorphisms of elliptic curves. In The Grothendieck Festschrift, Vol. I (Vol. 86, pp. 33–85). Birkhäuser Boston, Boston, MA.

MR1086882

Code

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

needsPackage "AssociativeAlgebras"
K = frac(QQ[q12, q13, q14, q23, q24, q34]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
  x1*x2 - q12*x2*x1,
  x1*x3 - q13*x3*x1,
  x1*x4 - q14*x4*x1,
  x2*x3 - q23*x3*x2,
  x2*x4 - q24*x4*x2,
  x3*x4 - q34*x4*x3
};
B = A/I;

PolyRing := FunctionField(Rationals, ["q12", "q13", "q14", "q23", "q24", "q34"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
q12 := indets[1];;
q13 := indets[2];;
q14 := indets[3];;
q23 := indets[4];;
q24 := indets[5];;
q34 := indets[6];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
  x1*x2 - q12*x2*x1,
  x1*x3 - q13*x3*x1,
  x1*x4 - q14*x4*x1,
  x2*x3 - q23*x3*x2,
  x2*x4 - q24*x4*x2,
  x3*x4 - q34*x4*x3
];;
A := kQ / rels;;

// untested (Magma not available here)
K<q12, q13, q14, q23, q24, q34> := RationalFunctionField(Rationals(), 6);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
  x1*x2 - q12*x2*x1,
  x1*x3 - q13*x3*x1,
  x1*x4 - q14*x4*x1,
  x2*x3 - q23*x3*x2,
  x2*x4 - q24*x4*x2,
  x3*x4 - q34*x4*x3
];
A := quo< F | rels >;

4d-AS-regular

skew

References

Code