Jordan

An "additive" deformation: the commutators of the generators are prescribed quadratic forms with coefficients polynomial in the single parameter \(a\), realising a quantization of a Poisson structure whose semiclassical limit has a \(\mathbb{G}_{\mathrm{a}}\)-symmetry. Its point modules are parametrised by a bouquet of rational normal curves (a line, a conic and a twisted cubic). It is called *Jordan* because it is related to the Jordan plane \(k\langle x, y\rangle/(yx - xy - x^2)\): quotienting repeatedly by the first variable \(x_1\) recovers the Jordan plane.

Relations

\((x_1 x_2 - x_2 x_1) + 4 a x_1^2\)
\((x_2 x_3 - x_3 x_2) + 4 (a+1) x_2^2 - 8 (a+1)(a+2) x_1 x_2 - 4 (a+2) x_1 x_3\)
\((x_1 x_3 - x_3 x_1) + 4 a x_1 x_2 + 8 a^2 x_1^2 - 8 a x_1^2\)
\((x_2 x_4 - x_4 x_2) + 4 (a+1) x_2 x_3 + 8 a (a+1) x_2^2 - (64/3) a (a+1)(a+2) x_1 x_2 - 16 (a+1)(a+2) x_1 x_3 - 4 (a+3) x_1 x_4\)
\((x_1 x_4 - x_4 x_1) + 4 a x_1 x_3 - 8 (a - a^2) x_1 x_2 - (64/6)(-a^3 + 3 a^2 - 2 a) x_1^2\)
\((x_3 x_4 - x_4 x_3) + 4 (a+2) x_3^2 - 8 (a+2)(a+3) x_2 x_3 + (64/6)(a+2)(a+3)(a+4) x_1 x_3 - 4 (a+3) x_2 x_4 + 8 (a+3)(a+4) x_1 x_4\)

Parameters

\(a\) — A single scalar parameter.

Point scheme

a line, a conic and a twisted cubic (a bouquet of rational normal curves) (dimension \(1\)).

Centre

trivial in low degrees (\(\dim \operatorname{Z}(A)_2 = \dim \operatorname{Z}(A)_3 = 0\)), as for a generic noncommutative \(\mathbb{P}^3\).

Normal elements

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

Hochschild cohomology of \(A\)

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

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

\(\mathrm{HH}^\bullet(\operatorname{qgr} A) = (1, 1, 0, 4)\)

Kodaira–Spencer

rank \(1\)
injective: yes
surjective: yes

Introduced

2016, MR3901632. A member of the additive (\(\mathbb{G}_{\mathrm{a}}\)) family of regular algebras of Lecoutre–Sierra, obtained as a quantization of a Poisson structure.

References

Lecoutre, C., & Sierra, S. J. (2019). A new family of Poisson algebras and their deformations. Nagoya Math. J., 233, 32–86.

MR3901632 doi

Code

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

needsPackage "AssociativeAlgebras"
K = frac(QQ[a]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
  (x1*x2 - x2*x1) + 4*a*x1^2,
  (x2*x3 - x3*x2) + 4*(a+1)*x2^2 - 8*(a+1)*(a+2)*x1*x2 - 4*(a+2)*x1*x3,
  (x1*x3 - x3*x1) + 4*a*x1*x2 + 8*a^2*x1^2 - 8*a*x1^2,
  (x2*x4 - x4*x2) + 4*(a+1)*x2*x3 + 8*a*(a+1)*x2^2 - (64/3)*a*(a+1)*(a+2)*x1*x2 - 16*(a+1)*(a+2)*x1*x3 - 4*(a+3)*x1*x4,
  (x1*x4 - x4*x1) + 4*a*x1*x3 - 8*(a - a^2)*x1*x2 - (64/6)*(-a^3 + 3*a^2 - 2*a)*x1^2,
  (x3*x4 - x4*x3) + 4*(a+2)*x3^2 - 8*(a+2)*(a+3)*x2*x3 + (64/6)*(a+2)*(a+3)*(a+4)*x1*x3 - 4*(a+3)*x2*x4 + 8*(a+3)*(a+4)*x1*x4
};
B = A/I;

PolyRing := FunctionField(Rationals, ["a"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
a := indets[1];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
  (x1*x2 - x2*x1) + 4*a*x1^2,
  (x2*x3 - x3*x2) + 4*(a+1)*x2^2 - 8*(a+1)*(a+2)*x1*x2 - 4*(a+2)*x1*x3,
  (x1*x3 - x3*x1) + 4*a*x1*x2 + 8*a^2*x1^2 - 8*a*x1^2,
  (x2*x4 - x4*x2) + 4*(a+1)*x2*x3 + 8*a*(a+1)*x2^2 - (64/3)*a*(a+1)*(a+2)*x1*x2 - 16*(a+1)*(a+2)*x1*x3 - 4*(a+3)*x1*x4,
  (x1*x4 - x4*x1) + 4*a*x1*x3 - 8*(a - a^2)*x1*x2 - (64/6)*(-a^3 + 3*a^2 - 2*a)*x1^2,
  (x3*x4 - x4*x3) + 4*(a+2)*x3^2 - 8*(a+2)*(a+3)*x2*x3 + (64/6)*(a+2)*(a+3)*(a+4)*x1*x3 - 4*(a+3)*x2*x4 + 8*(a+3)*(a+4)*x1*x4
];;
A := kQ / rels;;

// untested (Magma not available here)
K<a> := RationalFunctionField(Rationals(), 1);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
  (x1*x2 - x2*x1) + 4*a*x1^2,
  (x2*x3 - x3*x2) + 4*(a+1)*x2^2 - 8*(a+1)*(a+2)*x1*x2 - 4*(a+2)*x1*x3,
  (x1*x3 - x3*x1) + 4*a*x1*x2 + 8*a^2*x1^2 - 8*a*x1^2,
  (x2*x4 - x4*x2) + 4*(a+1)*x2*x3 + 8*a*(a+1)*x2^2 - (64/3)*a*(a+1)*(a+2)*x1*x2 - 16*(a+1)*(a+2)*x1*x3 - 4*(a+3)*x1*x4,
  (x1*x4 - x4*x1) + 4*a*x1*x3 - 8*(a - a^2)*x1*x2 - (64/6)*(-a^3 + 3*a^2 - 2*a)*x1^2,
  (x3*x4 - x4*x3) + 4*(a+2)*x3^2 - 8*(a+2)*(a+3)*x2*x3 + (64/6)*(a+2)*(a+3)*(a+4)*x1*x3 - 4*(a+3)*x2*x4 + 8*(a+3)*(a+4)*x1*x4
];
A := quo< F | rels >;

4d-AS-regular

Jordan

References

Code