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The sea of Poisson Structures and its Graph Complex flows

Index » Generic Poisson structure on $\mathbb{R}^2$ » Deformation from 3-wheel graph cohomology class

Poisson structure:	Generic Poisson structure on $\mathbb{R}^2$
Graph cohomology class:	3-wheel graph cohomology class
Tags:	-
References:	[`Kontsevich1996`, section 5.3]: Claim that $Q = [P,X]$. [`BouisaghouaneBuringKiselev2016`, Remark 13]: Expression for $X$ in terms of directed graphs. [`Bouisaghouane2017`]: Expressions for $Q, X$, and $H$ such that $X = [\xi_1\xi_2, H]$. Also $X$ and $H$ in terms of directed graphs. [`BuringPhD2022`, chapter I.6.1]: Code to compute $Q, X$ and $H$. Also $H$ in terms of directed graphs.

Bi-vector field deformation term $Q$ with $[P,Q] = 0$

Formula for $Q$:

(8*u_y^3*u_xxx - 24*u_x*u_y^2*u_xxy + 24*u_x^2*u_y*u_xyy - 8*u_x^3*u_yyy)*xi1*xi2

Is a Poisson 2-coboundary: Yes, a vector field $X$ such that $Q=[P,X]$ is given below.

Code (gcaops):

from gcaops.all import *

D = DifferentialPolynomialRing(QQ, fibre_names=['u'], base_names=['x','y'], max_differential_orders=[3])
u = D.fibre_variables()[0]
x, y = D.base_variables()

S.<xi1,xi2> = SuperfunctionAlgebra(D, (x, y))
P = u*xi1*xi2

GC = UndirectedGraphComplex(QQ, implementation='vector', sparse=True)
gamma3 = GC.cohomology_basis(4,6)[0]

Q_gamma3_2d = S.zero()
try:
    Q_gamma3_2d = load('Q_gamma3_2d.sobj')
except FileNotFoundError:
    gamma3_op_2d = S.graph_operation(gamma3)
    Q_gamma3_2d = gamma3_op_2d(P,P,P,P)
    save(Q_gamma3_2d, 'Q_gamma3_2d.sobj')

with open('Q_gamma3_2d.txt', 'w') as f:
    print(Q_gamma3_2d, file=f)

Vector field $X$ such that $Q = [P,X]$

Formula for $X$:

(-16*u_y*u_xy^2 + 16*u_y*u_xx*u_yy + 8*u_y^2*u_xxy - 16*u_x*u_y*u_xyy + 8*u_x^2*u_yyy)*xi1 + (16*u_x*u_xy^2 - 16*u_x*u_xx*u_yy - 8*u_y^2*u_xxx + 16*u_x*u_y*u_xxy - 8*u_x^2*u_xyy)*xi2

Hamiltonian function $H$ such that $X = [\xi_1\xi_2, H]$

Formula for $H$:

8*u_y^2*u_xx - 16*u_x*u_y*u_xy + 8*u_x^2*u_yy