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The sea of Poisson Structures and its Graph Complex flows

Index » Generic Poisson structure on $\mathbb{R}^2$ » Deformation from 5-wheel graph cohomology class

Poisson structure:Generic Poisson structure on $\mathbb{R}^2$
Graph cohomology class:5-wheel graph cohomology class
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References:

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

Formula for $Q$:
(-10*u_y^3*u_xx*u_yy*u_xxx + 20*u_x*u_y^2*u_xy*u_yy*u_xxx - 10*u_x^2*u_y*u_yy^2*u_xxx + 20*u_y^3*u_xx*u_xy*u_xxy - 40*u_x*u_y^2*u_xy^2*u_xxy + 10*u_x*u_y^2*u_xx*u_yy*u_xxy + 10*u_x^3*u_yy^2*u_xxy - 10*u_y^3*u_xx^2*u_xyy + 40*u_x^2*u_y*u_xy^2*u_xyy - 10*u_x^2*u_y*u_xx*u_yy*u_xyy - 20*u_x^3*u_xy*u_yy*u_xyy + 10*u_x*u_y^2*u_xx^2*u_yyy - 20*u_x^2*u_y*u_xx*u_xy*u_yyy + 10*u_x^3*u_xx*u_yy*u_yyy - 10*u_y^4*u_xy*u_xxxx + 10*u_x*u_y^3*u_yy*u_xxxx + 10*u_y^4*u_xx*u_xxxy + 20*u_x*u_y^3*u_xy*u_xxxy - 30*u_x^2*u_y^2*u_yy*u_xxxy - 30*u_x*u_y^3*u_xx*u_xxyy + 30*u_x^3*u_y*u_yy*u_xxyy + 30*u_x^2*u_y^2*u_xx*u_xyyy - 20*u_x^3*u_y*u_xy*u_xyyy - 10*u_x^4*u_yy*u_xyyy - 10*u_x^3*u_y*u_xx*u_yyyy + 10*u_x^4*u_xy*u_yyyy - 2*u_y^5*u_xxxxx + 10*u_x*u_y^4*u_xxxxy - 20*u_x^2*u_y^3*u_xxxyy + 20*u_x^3*u_y^2*u_xxyyy - 10*u_x^4*u_y*u_xyyyy + 2*u_x^5*u_yyyyy)*xi1*xi2
Is a Poisson 2-coboundary:?
Code (gcaops):
from gcaops.all import *

D = DifferentialPolynomialRing(QQ, fibre_names=['u'], base_names=['x','y'], max_differential_orders=[5])
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)
gamma5 = GC.cohomology_basis(6,10)[0]

Q_gamma5_2d = S.zero()
try:
    Q_gamma5_2d = load('Q_gamma5_2d.sobj')
except FileNotFoundError:
    gamma5_op_2d = S.graph_operation(gamma5)
    Q_gamma5_2d = gamma5_op_2d(P,P,P,P,P,P)
    save(Q_gamma5_2d, 'Q_gamma5_2d.sobj')

with open('Q_gamma5_2d.txt', 'w') as f:
    print(Q_gamma5_2d, file=f)