Cassidy–Goetz–Shelton

The Cassidy–Goetz–Shelton algebra: \(x_1, x_2\) commute, the pairs involving \(x_3, x_4\) \(q\)-commute, and the \(x_3 x_4\) relation carries a quadratic correction in \(x_1, x_2\). The \(σ\) is a diagonal twist (the parameters \(α, β\)), under which the family is closed; the \(σ\)-twisted form is the one appearing in the dimension-four classification.

Arises as a quantization of a quadratic Poisson structure on \(\mathbb{P}^3\) (Pym's classification).

Relations

\(α (x_2 x_1 - x_1 x_2)\)
\(β x_3 x_1 - (1/p_0) α x_1 x_3\)
\((α^2/β) x_4 x_1 - p_0 α x_1 x_4\)
\(β x_3 x_2 - p_1 α x_2 x_3\)
\((α^2/β) x_4 x_2 - (1/p_1) α x_2 x_4\)
\((α^2/β) x_4 x_3 - p_1 (1/p_0) β x_3 x_4 - α (p_1 - p_0)(x_1^2 + λ x_1 x_2 + x_2^2) - α (1 - p_0^2) x_1^2 - α (p_1^2 - 1) x_2^2\)

Parameters

\(p_0, p_1, λ, α, β\) — \(p_0, p_1\) nonzero scalars; \(λ, α, β\) scalars (\(α, β\) rescaling the \(σ\)-twist).

Point scheme

a line, two conics and two points (dimension \(1\)).

Centre

\(\dim \operatorname{Z}(A)_2 = 0\), \(\dim \operatorname{Z}(A)_3 = 0\), \(\dim \operatorname{Z}(A)_4 = 0\).

Normal elements

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

Hochschild cohomology of \(A\)

\(\mathrm{HH}^\bullet_0(A) = (1, 2, 4, 8)\)

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

\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 1, 3, 7)\)

Kodaira–Spencer

rank \(4\)
injective: no
surjective: yes

Introduced

2006, MR2253666. A generalized Laurent polynomial ring of Cassidy–Goetz–Shelton; in Pym's classification of quadratic Poisson structures it is the type \(L(1,1,2)\).

References

Cassidy, T., Goetz, P., & Shelton, B. (2006). Generalized Laurent polynomial rings as quantum projective 3-spaces. J. Algebra, 303(1), 358–372.

MR2253666 doi

Pym, B. (2015). Quantum deformations of projective three-space. Adv. Math., 281, 1216–1241.

MR3366864 doi

Code

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

needsPackage "AssociativeAlgebras"
K = frac(QQ[alpha, beta, lambda, p0, p1]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
  alpha*(x2*x1 - x1*x2),
  beta*x3*x1 - (1/p0)*alpha*x1*x3,
  (alpha^2/beta)*x4*x1 - p0*alpha*x1*x4,
  beta*x3*x2 - p1*alpha*x2*x3,
  (alpha^2/beta)*x4*x2 - (1/p1)*alpha*x2*x4,
  (alpha^2/beta)*x4*x3 - p1*(1/p0)*beta*x3*x4 - alpha*(p1 - p0)*(x1^2 + lambda*x1*x2 + x2^2) - alpha*(1 - p0^2)*x1^2 - alpha*(p1^2 - 1)*x2^2
};
B = A/I;

PolyRing := FunctionField(Rationals, ["alpha", "beta", "lambda", "p0", "p1"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
alpha := indets[1];;
beta := indets[2];;
lambda := indets[3];;
p0 := indets[4];;
p1 := indets[5];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
  alpha*(x2*x1 - x1*x2),
  beta*x3*x1 - (1/p0)*alpha*x1*x3,
  (alpha^2/beta)*x4*x1 - p0*alpha*x1*x4,
  beta*x3*x2 - p1*alpha*x2*x3,
  (alpha^2/beta)*x4*x2 - (1/p1)*alpha*x2*x4,
  (alpha^2/beta)*x4*x3 - p1*(1/p0)*beta*x3*x4 - alpha*(p1 - p0)*(x1^2 + lambda*x1*x2 + x2^2) - alpha*(1 - p0^2)*x1^2 - alpha*(p1^2 - 1)*x2^2
];;
A := kQ / rels;;

// untested (Magma not available here)
K<alpha, beta, lambda, p0, p1> := RationalFunctionField(Rationals(), 5);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
  alpha*(x2*x1 - x1*x2),
  beta*x3*x1 - (1/p0)*alpha*x1*x3,
  (alpha^2/beta)*x4*x1 - p0*alpha*x1*x4,
  beta*x3*x2 - p1*alpha*x2*x3,
  (alpha^2/beta)*x4*x2 - (1/p1)*alpha*x2*x4,
  (alpha^2/beta)*x4*x3 - p1*(1/p0)*beta*x3*x4 - alpha*(p1 - p0)*(x1^2 + lambda*x1*x2 + x2^2) - alpha*(1 - p0^2)*x1^2 - alpha*(p1^2 - 1)*x2^2
];
A := quo< F | rels >;

4d-AS-regular

Cassidy–Goetz–Shelton

References

Code