Show code: [gcaops]

The sea of Poisson Structures and its Graph Complex flows

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

Poisson structure:	Generic Poisson structure on $\mathbb{R}^3$
Graph cohomology class:	3-wheel graph cohomology class
Tags:	-
References:	-

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

Formula for $Q$:

(24*u_zz*v_z^2*v_xx - 48*u_z*u_zz*v_z*v_xy + 48*u_z*u_zz*v_y*v_xz - 48*u_zz*v_x*v_z*v_xz + 24*u_z^2*u_zz*v_yy - 48*u_y*u_z*u_zz*v_yz + 48*u_x*u_zz*v_z*v_yz - 24*u_z^2*u_yy*v_zz + 48*u_y*u_z*u_yz*v_zz - 48*u_z*u_xz*v_y*v_zz + 48*u_z*u_xy*v_z*v_zz - 48*u_x*u_yz*v_z*v_zz + 48*u_xz*v_x*v_z*v_zz - 24*u_xx*v_z^2*v_zz + 48*u_yz*v_z*v_xz*w_x - 48*u_xz*v_z*v_yz*w_x - 48*u_z*u_yz*v_xz*w_y + 48*u_z*u_xz*v_yz*w_y - 24*u_zz*v_y*v_xx*w_z - 24*u_yz*v_z*v_xx*w_z + 24*u_z*u_yz*v_xy*w_z + 24*u_y*u_zz*v_xy*w_z + 24*u_zz*v_x*v_xy*w_z + 24*u_xz*v_z*v_xy*w_z + 24*u_z*u_yy*v_xz*w_z - 24*u_y*u_yz*v_xz*w_z + 24*u_yz*v_x*v_xz*w_z - 24*u_xy*v_z*v_xz*w_z - 24*u_z*u_xz*v_yy*w_z - 24*u_x*u_zz*v_yy*w_z - 24*u_z*u_xy*v_yz*w_z + 24*u_y*u_xz*v_yz*w_z - 24*u_xz*v_x*v_yz*w_z + 24*u_xx*v_z*v_yz*w_z - 24*u_y*u_xy*v_zz*w_z + 24*u_x*u_yy*v_zz*w_z - 24*u_xy*v_x*v_zz*w_z + 24*u_xx*v_y*v_zz*w_z - 24*u_yz*v_xy*w_x*w_z - 24*u_yy*v_xz*w_x*w_z + 24*u_xz*v_yy*w_x*w_z + 24*u_xy*v_yz*w_x*w_z + 24*u_yz*v_xx*w_y*w_z - 24*u_xz*v_xy*w_y*w_z + 24*u_xy*v_xz*w_y*w_z - 24*u_xx*v_yz*w_y*w_z - 24*u_yz*v_z^2*w_xx + 24*v_z^2*v_xz*w_xx + 24*u_yz*v_y*w_z*w_xx + 24*u_yy*v_z*w_z*w_xx - 24*v_z*v_xy*w_z*w_xx - 24*v_y*v_xz*w_z*w_xx - 24*u_yy*w_y*w_z*w_xx + 24*v_xy*w_y*w_z*w_xx + 48*u_z*u_yz*v_z*w_xy - 48*u_z*v_z*v_xz*w_xy - 24*u_z*u_yy*w_z*w_xy - 24*u_y*u_yz*w_z*w_xy - 24*u_yz*v_x*w_z*w_xy - 24*u_xy*v_z*w_z*w_xy + 24*v_z*v_xx*w_z*w_xy + 24*u_z*v_xy*w_z*w_xy + 24*u_y*v_xz*w_z*w_xy + 24*v_x*v_xz*w_z*w_xy + 24*u_yy*w_x*w_z*w_xy - 24*v_xy*w_x*w_z*w_xy + 24*u_xy*w_y*w_z*w_xy - 24*v_xx*w_y*w_z*w_xy - 24*u_y*w_z*w_xy^2 + 24*v_x*w_z*w_xy^2 - 24*u_z*u_yz*v_y*w_xz - 24*u_y*u_zz*v_y*w_xz + 24*u_zz*v_x*v_y*w_xz - 24*u_z*u_yy*v_z*w_xz + 24*u_y*u_yz*v_z*w_xz + 24*u_yz*v_x*v_z*w_xz - 24*u_xz*v_y*v_z*w_xz + 24*u_xy*v_z^2*w_xz - 24*v_z^2*v_xx*w_xz + 24*u_z*v_z*v_xy*w_xz + 24*u_z*v_y*v_xz*w_xz - 24*u_x*v_z*v_yz*w_xz - 24*v_x^2*v_zz*w_xz + 24*u_x*v_y*v_zz*w_xz - 24*u_yz*v_y*w_x*w_xz - 24*u_yy*v_z*w_x*w_xz + 24*v_z*v_xy*w_x*w_xz - 24*v_y*v_xz*w_x*w_xz + 48*v_x*v_yz*w_x*w_xz - 24*v_yy*w_x^2*w_xz + 24*u_z*u_yy*w_y*w_xz + 24*u_y*u_yz*w_y*w_xz - 24*u_yz*v_x*w_y*w_xz + 24*u_xz*v_y*w_y*w_xz - 24*u_z*v_xy*w_y*w_xz - 24*u_x*v_yz*w_y*w_xz + 24*u_yy*w_x*w_y*w_xz + 24*v_xy*w_x*w_y*w_xz - 24*u_xy*w_y^2*w_xz + 24*v_y*v_xx*w_z*w_xz - 24*u_y*v_xy*w_z*w_xz - 24*v_x*v_xy*w_z*w_xz + 24*u_x*v_yy*w_z*w_xz + 24*u_y*v_z*w_xy*w_xz - 24*v_x*v_z*w_xy*w_xz + 24*u_y*w_y*w_xy*w_xz - 24*v_x*w_y*w_xy*w_xz - 24*u_y*v_y*w_xz^2 + 24*v_x*v_y*w_xz^2 - 24*u_z^2*u_yz*w_yy + 24*u_z^2*v_xz*w_yy + 24*u_z*u_xy*w_z*w_yy + 24*u_x*u_yz*w_z*w_yy - 24*u_z*v_xx*w_z*w_yy - 24*u_x*v_xz*w_z*w_yy - 24*u_xy*w_x*w_z*w_yy + 24*v_xx*w_x*w_z*w_yy + 24*u_y*w_z*w_xx*w_yy - 24*v_x*w_z*w_xx*w_yy - 24*u_y*u_z*w_xz*w_yy + 24*u_z*v_x*w_xz*w_yy - 24*u_y*w_x*w_xz*w_yy + 24*v_x*w_x*w_xz*w_yy + 24*u_z^2*u_yy*w_yz + 24*u_y^2*u_zz*w_yz + 24*u_z*u_xz*v_y*w_yz - 24*u_x*u_zz*v_y*w_yz - 24*u_z*u_xy*v_z*w_yz - 24*u_x*u_yz*v_z*w_yz + 24*u_z*v_z*v_xx*w_yz - 24*u_z^2*v_xy*w_yz - 24*u_y*u_z*v_xz*w_yz - 24*u_z*v_x*v_xz*w_yz + 24*u_x*v_z*v_xz*w_yz + 24*u_x*u_z*v_yz*w_yz - 24*u_x*u_y*v_zz*w_yz + 24*u_x*v_x*v_zz*w_yz + 24*u_xz*v_y*w_x*w_yz + 24*u_xy*v_z*w_x*w_yz - 24*v_z*v_xx*w_x*w_yz + 24*u_y*v_xz*w_x*w_yz - 24*v_x*v_xz*w_x*w_yz - 24*u_x*v_yz*w_x*w_yz + 24*v_xy*w_x^2*w_yz - 24*u_z*u_xy*w_y*w_yz - 48*u_y*u_xz*w_y*w_yz + 24*u_x*u_yz*w_y*w_yz + 24*u_z*v_xx*w_y*w_yz + 24*u_x*v_xz*w_y*w_yz - 24*u_xy*w_x*w_y*w_yz - 24*v_xx*w_x*w_y*w_yz + 24*u_xx*w_y^2*w_yz + 24*u_y*u_xy*w_z*w_yz - 24*u_x*u_yy*w_z*w_yz + 24*u_xy*v_x*w_z*w_yz - 24*u_xx*v_y*w_z*w_yz - 24*u_y*v_z*w_xx*w_yz + 24*v_x*v_z*w_xx*w_yz - 24*u_y*w_y*w_xx*w_yz + 24*v_x*w_y*w_xx*w_yz + 24*u_y*u_z*w_xy*w_yz - 24*u_z*v_x*w_xy*w_yz + 24*u_y*w_x*w_xy*w_yz - 24*v_x*w_x*w_xy*w_yz + 24*u_y^2*w_xz*w_yz - 24*v_x^2*w_xz*w_yz - 24*u_x*u_y*w_yz^2 + 24*u_x*v_x*w_yz^2 - 24*u_y^2*u_yz*w_zz + 24*u_y*u_xz*v_y*w_zz + 24*u_x*u_yz*v_y*w_zz - 24*u_xz*v_x*v_y*w_zz - 24*u_y*u_xy*v_z*w_zz + 24*u_x*u_yy*v_z*w_zz - 24*u_xy*v_x*v_z*w_zz + 24*u_xx*v_y*v_z*w_zz - 24*u_z*v_y*v_xx*w_zz + 24*u_y*u_z*v_xy*w_zz + 24*u_z*v_x*v_xy*w_zz + 24*v_x^2*v_xz*w_zz - 24*u_x*v_y*v_xz*w_zz - 24*u_x*u_z*v_yy*w_zz + 24*u_x*u_y*v_yz*w_zz - 24*u_x*v_x*v_yz*w_zz + 24*v_y*v_xx*w_x*w_zz - 24*u_y*v_xy*w_x*w_zz - 24*v_x*v_xy*w_x*w_zz + 24*u_x*v_yy*w_x*w_zz + 24*u_y*u_xy*w_y*w_zz - 24*u_x*u_yy*w_y*w_zz + 24*u_xy*v_x*w_y*w_zz - 24*u_xx*v_y*w_y*w_zz + 24*u_y*v_y*w_xx*w_zz - 24*v_x*v_y*w_xx*w_zz - 24*u_y^2*w_xy*w_zz + 24*v_x^2*w_xy*w_zz + 24*u_x*u_y*w_yy*w_zz - 24*u_x*v_x*w_yy*w_zz - 8*v_z^3*w_xxx + 8*w_y^3*w_xxx + 24*v_y*v_z*w_z*w_xxx - 24*v_y*w_y*w_z*w_xxx + 24*u_z*v_z^2*w_xxy - 24*w_x*w_y^2*w_xxy - 24*u_z*v_y*w_z*w_xxy - 24*u_y*v_z*w_z*w_xxy - 24*v_x*v_z*w_z*w_xxy + 24*v_y*w_x*w_z*w_xxy + 24*u_y*w_y*w_z*w_xxy + 24*v_x*w_y*w_z*w_xxy - 24*u_z*v_y*v_z*w_xxz + 24*v_x*v_z^2*w_xxz - 24*v_y*v_z*w_x*w_xxz + 24*u_z*v_y*w_y*w_xxz + 24*v_y*w_x*w_y*w_xxz - 24*u_y*w_y^2*w_xxz + 24*u_y*v_y*w_z*w_xxz - 24*v_x*v_y*w_z*w_xxz - 24*u_z^2*v_z*w_xyy + 24*w_x^2*w_y*w_xyy + 24*u_y*u_z*w_z*w_xyy + 24*u_z*v_x*w_z*w_xyy + 24*u_x*v_z*w_z*w_xyy - 24*u_y*w_x*w_z*w_xyy - 24*v_x*w_x*w_z*w_xyy - 24*u_x*w_y*w_z*w_xyy + 24*u_z^2*v_y*w_xyz + 24*u_y*u_z*v_z*w_xyz - 24*u_z*v_x*v_z*w_xyz - 24*u_x*v_z^2*w_xyz + 24*u_y*v_z*w_x*w_xyz + 24*v_x*v_z*w_x*w_xyz - 24*v_y*w_x^2*w_xyz - 24*u_y*u_z*w_y*w_xyz - 24*u_z*v_x*w_y*w_xyz + 24*u_y*w_x*w_y*w_xyz - 24*v_x*w_x*w_y*w_xyz + 24*u_x*w_y^2*w_xyz - 24*u_y^2*w_z*w_xyz + 24*v_x^2*w_z*w_xyz - 24*u_y*u_z*v_y*w_xzz + 24*u_z*v_x*v_y*w_xzz - 24*v_x^2*v_z*w_xzz + 24*u_x*v_y*v_z*w_xzz - 24*u_y*v_y*w_x*w_xzz + 24*v_x*v_y*w_x*w_xzz + 24*u_y^2*w_y*w_xzz - 24*u_x*v_y*w_y*w_xzz + 8*u_z^3*w_yyy - 8*w_x^3*w_yyy - 24*u_x*u_z*w_z*w_yyy + 24*u_x*w_x*w_z*w_yyy - 24*u_y*u_z^2*w_yyz + 24*u_x*u_z*v_z*w_yyz - 24*u_x*v_z*w_x*w_yyz + 24*v_x*w_x^2*w_yyz + 24*u_x*u_z*w_y*w_yyz - 24*u_x*w_x*w_y*w_yyz + 24*u_x*u_y*w_z*w_yyz - 24*u_x*v_x*w_z*w_yyz + 24*u_y^2*u_z*w_yzz - 24*u_x*u_z*v_y*w_yzz - 24*u_x*u_y*v_z*w_yzz + 24*u_x*v_x*v_z*w_yzz - 24*v_x^2*w_x*w_yzz + 24*u_x*v_y*w_x*w_yzz - 24*u_x*u_y*w_y*w_yzz + 24*u_x*v_x*w_y*w_yzz - 8*u_y^3*w_zzz + 8*v_x^3*w_zzz + 24*u_x*u_y*v_y*w_zzz - 24*u_x*v_x*v_y*w_zzz)*xi1*xi2 + (-24*u_zz*v_y*v_z*v_xx + 24*u_z*u_yz*v_z*v_xy + 24*u_y*u_zz*v_z*v_xy + 24*u_zz*v_x*v_z*v_xy - 24*u_xz*v_z^2*v_xy - 24*u_z*u_yz*v_y*v_xz - 24*u_y*u_zz*v_y*v_xz + 24*u_zz*v_x*v_y*v_xz + 24*u_xz*v_y*v_z*v_xz + 24*u_z*v_z*v_xy*v_xz - 24*u_z*v_y*v_xz^2 - 24*u_z^2*u_yz*v_yy + 24*u_z*u_xz*v_z*v_yy - 24*u_x*u_zz*v_z*v_yy - 24*u_z^2*v_xz*v_yy + 24*u_z^2*u_yy*v_yz + 24*u_y^2*u_zz*v_yz + 24*u_z*u_xz*v_y*v_yz - 24*u_x*u_zz*v_y*v_yz - 48*u_z*u_xy*v_z*v_yz - 24*u_y*u_xz*v_z*v_yz + 24*u_x*u_yz*v_z*v_yz - 24*u_xz*v_x*v_z*v_yz + 24*u_xx*v_z^2*v_yz - 24*u_z*v_z*v_xx*v_yz + 24*u_z^2*v_xy*v_yz + 24*u_y*u_z*v_xz*v_yz + 24*u_z*v_x*v_xz*v_yz - 24*u_x*u_z*v_yz^2 - 24*u_y^2*u_yz*v_zz + 24*u_y*u_xz*v_y*v_zz + 24*u_x*u_yz*v_y*v_zz - 24*u_xz*v_x*v_y*v_zz + 24*u_z*v_y*v_xx*v_zz - 24*u_y*u_z*v_xy*v_zz - 24*u_z*v_x*v_xy*v_zz + 24*u_x*u_z*v_yy*v_zz + 8*v_z^3*v_xxx - 24*u_z*v_z^2*v_xxy + 24*u_z*v_y*v_z*v_xxz - 24*v_x*v_z^2*v_xxz + 24*u_z^2*v_z*v_xyy - 24*u_z^2*v_y*v_xyz - 24*u_y*u_z*v_z*v_xyz + 24*u_z*v_x*v_z*v_xyz + 24*u_x*v_z^2*v_xyz + 24*u_y*u_z*v_y*v_xzz - 24*u_z*v_x*v_y*v_xzz + 24*v_x^2*v_z*v_xzz - 24*u_x*v_y*v_z*v_xzz - 8*u_z^3*v_yyy + 24*u_y*u_z^2*v_yyz - 24*u_x*u_z*v_z*v_yyz - 24*u_y^2*u_z*v_yzz + 24*u_x*u_z*v_y*v_yzz + 24*u_x*u_y*v_z*v_yzz - 24*u_x*v_x*v_z*v_yzz + 8*u_y^3*v_zzz - 8*v_x^3*v_zzz - 24*u_x*u_y*v_y*v_zzz + 24*u_x*v_x*v_y*v_zzz - 24*u_yz*v_z*v_xy*w_x - 24*u_yz*v_y*v_xz*w_x - 24*v_z*v_xy*v_xz*w_x + 24*v_y*v_xz^2*w_x + 24*u_xz*v_z*v_yy*w_x + 24*u_xz*v_y*v_yz*w_x + 24*v_z*v_xx*v_yz*w_x - 24*u_y*v_xz*v_yz*w_x - 24*v_x*v_xz*v_yz*w_x + 24*u_x*v_yz^2*w_x - 24*v_y*v_xx*v_zz*w_x + 24*u_y*v_xy*v_zz*w_x + 24*v_x*v_xy*v_zz*w_x - 24*u_x*v_yy*v_zz*w_x + 24*v_y*v_z*v_xxz*w_x - 24*u_y*v_z*v_xyz*w_x - 24*v_x*v_z*v_xyz*w_x + 24*u_y*v_y*v_xzz*w_x - 24*v_x*v_y*v_xzz*w_x + 24*u_x*v_z*v_yyz*w_x + 24*v_x^2*v_yzz*w_x - 24*u_x*v_y*v_yzz*w_x + 24*v_xz*v_yy*w_x^2 - 24*v_xy*v_yz*w_x^2 + 24*v_y*v_xyz*w_x^2 - 24*v_x*v_yyz*w_x^2 + 8*v_yyy*w_x^3 + 24*u_zz*v_y*v_xx*w_y + 24*u_z*u_yz*v_xy*w_y - 24*u_y*u_zz*v_xy*w_y - 24*u_zz*v_x*v_xy*w_y + 48*u_y*u_yz*v_xz*w_y - 24*u_xz*v_y*v_xz*w_y + 24*u_z*v_xy*v_xz*w_y - 24*u_z*u_xz*v_yy*w_y + 24*u_x*u_zz*v_yy*w_y - 24*u_y*u_xz*v_yz*w_y - 24*u_x*u_yz*v_yz*w_y + 24*u_xz*v_x*v_yz*w_y - 24*u_z*v_xx*v_yz*w_y - 24*u_z*v_y*v_xxz*w_y + 24*u_y*u_z*v_xyz*w_y + 24*u_z*v_x*v_xyz*w_y - 24*u_y^2*v_xzz*w_y + 24*u_x*v_y*v_xzz*w_y - 24*u_x*u_z*v_yyz*w_y + 24*u_x*u_y*v_yzz*w_y - 24*u_x*v_x*v_yzz*w_y + 24*u_yz*v_xy*w_x*w_y - 24*v_xy*v_xz*w_x*w_y - 24*u_xz*v_yy*w_x*w_y + 24*v_xx*v_yz*w_x*w_y - 24*v_y*v_xxz*w_x*w_y - 24*u_y*v_xyz*w_x*w_y + 24*v_x*v_xyz*w_x*w_y + 24*u_x*v_yyz*w_x*w_y - 24*v_xyy*w_x^2*w_y - 24*u_yz*v_xx*w_y^2 + 24*u_xz*v_xy*w_y^2 + 24*u_y*v_xxz*w_y^2 - 24*u_x*v_xyz*w_y^2 + 24*v_xxy*w_x*w_y^2 - 8*v_xxx*w_y^3 + 24*u_yz*v_y*v_xx*w_z - 24*u_z*u_yy*v_xy*w_z - 24*u_y*u_yz*v_xy*w_z - 24*u_yz*v_x*v_xy*w_z + 24*u_xy*v_z*v_xy*w_z - 24*u_z*v_xy^2*w_z + 24*u_z*u_xy*v_yy*w_z + 24*u_x*u_yz*v_yy*w_z - 24*u_xx*v_z*v_yy*w_z + 24*u_z*v_xx*v_yy*w_z + 24*u_y*u_xy*v_yz*w_z - 24*u_x*u_yy*v_yz*w_z + 24*u_xy*v_x*v_yz*w_z - 24*u_xx*v_y*v_yz*w_z - 24*v_y*v_z*v_xxx*w_z + 24*u_z*v_y*v_xxy*w_z + 24*u_y*v_z*v_xxy*w_z + 24*v_x*v_z*v_xxy*w_z - 24*u_y*v_y*v_xxz*w_z + 24*v_x*v_y*v_xxz*w_z - 24*u_y*u_z*v_xyy*w_z - 24*u_z*v_x*v_xyy*w_z - 24*u_x*v_z*v_xyy*w_z + 24*u_y^2*v_xyz*w_z - 24*v_x^2*v_xyz*w_z + 24*u_x*u_z*v_yyy*w_z - 24*u_x*u_y*v_yyz*w_z + 24*u_x*v_x*v_yyz*w_z + 24*u_yy*v_xy*w_x*w_z + 24*v_xy^2*w_x*w_z - 24*u_xy*v_yy*w_x*w_z - 24*v_xx*v_yy*w_x*w_z - 24*v_y*v_xxy*w_x*w_z + 24*u_y*v_xyy*w_x*w_z + 24*v_x*v_xyy*w_x*w_z - 24*u_x*v_yyy*w_x*w_z - 24*u_xy*v_xy*w_y*w_z + 24*u_xx*v_yy*w_y*w_z + 24*v_y*v_xxx*w_y*w_z - 24*u_y*v_xxy*w_y*w_z - 24*v_x*v_xxy*w_y*w_z + 24*u_x*v_xyy*w_y*w_z + 24*u_yz*v_y*v_z*w_xx - 24*v_y*v_z*v_xz*w_xx + 24*u_y*v_z*v_yz*w_xx - 24*v_x*v_z*v_yz*w_xx - 24*u_y*v_y*v_zz*w_xx + 24*v_x*v_y*v_zz*w_xx - 24*u_yz*v_y*w_y*w_xx + 24*v_y*v_xz*w_y*w_xx + 24*u_y*v_yz*w_y*w_xx - 24*v_x*v_yz*w_y*w_xx + 24*u_yy*w_y^2*w_xx - 24*v_xy*w_y^2*w_xx - 24*u_yy*v_y*w_z*w_xx + 24*v_y*v_xy*w_z*w_xx - 24*u_y*v_yy*w_z*w_xx + 24*v_x*v_yy*w_z*w_xx - 24*u_z*u_yz*v_y*w_xy + 24*u_y*u_zz*v_y*w_xy - 24*u_zz*v_x*v_y*w_xy - 48*u_y*u_yz*v_z*w_xy + 24*u_xz*v_y*v_z*w_xy + 24*u_z*v_y*v_xz*w_xy - 24*u_y*u_z*v_yz*w_xy + 24*u_z*v_x*v_yz*w_xy + 24*u_x*v_z*v_yz*w_xy + 24*u_y^2*v_zz*w_xy - 24*u_x*v_y*v_zz*w_xy + 24*u_yz*v_y*w_x*w_xy + 24*v_y*v_xz*w_x*w_xy - 24*u_y*v_yz*w_x*w_xy - 24*v_x*v_yz*w_x*w_xy + 24*v_yy*w_x^2*w_xy + 48*u_yz*v_x*w_y*w_xy - 24*u_xz*v_y*w_y*w_xy - 48*u_y*v_xz*w_y*w_xy + 24*u_x*v_yz*w_y*w_xy - 48*u_yy*w_x*w_y*w_xy + 24*v_xx*w_y^2*w_xy + 48*u_y*u_yy*w_z*w_xy - 24*v_y*v_xx*w_z*w_xy + 24*u_y*v_xy*w_z*w_xy - 24*v_x*v_xy*w_z*w_xy - 24*u_x*v_yy*w_z*w_xy + 24*u_z*u_yy*v_y*w_xz + 24*u_y*u_yz*v_y*w_xz - 24*u_yz*v_x*v_y*w_xz - 24*u_xy*v_y*v_z*w_xz + 24*v_y*v_z*v_xx*w_xz - 24*u_z*v_y*v_xy*w_xz - 24*u_y*v_z*v_xy*w_xz + 24*v_x*v_z*v_xy*w_xz + 24*u_y*v_y*v_xz*w_xz - 24*v_x*v_y*v_xz*w_xz + 24*u_y*u_z*v_yy*w_xz - 24*u_z*v_x*v_yy*w_xz - 24*u_y^2*v_yz*w_xz + 24*v_x^2*v_yz*w_xz + 24*u_yy*v_y*w_x*w_xz - 24*v_y*v_xy*w_x*w_xz + 24*u_y*v_yy*w_x*w_xz - 24*v_x*v_yy*w_x*w_xz - 48*u_y*u_yy*w_y*w_xz + 24*u_xy*v_y*w_y*w_xz - 24*v_y*v_xx*w_y*w_xz + 24*u_y*v_xy*w_y*w_xz + 24*v_x*v_xy*w_y*w_xz + 48*u_y*u_z*u_yz*w_yy - 24*u_y^2*u_zz*w_yy - 24*u_z*u_xz*v_y*w_yy + 24*u_x*u_zz*v_y*w_yy - 24*u_x*u_z*v_yz*w_yy - 24*u_xz*v_y*w_x*w_yy + 24*u_x*v_yz*w_x*w_yy - 24*v_xy*w_x^2*w_yy + 48*u_y*u_xz*w_y*w_yy - 48*u_x*u_yz*w_y*w_yy + 48*u_xy*w_x*w_y*w_yy - 24*u_xx*w_y^2*w_yy - 48*u_y*u_xy*w_z*w_yy + 24*u_xx*v_y*w_z*w_yy + 24*u_x*v_xy*w_z*w_yy - 48*u_y*u_z*u_yy*w_yz + 24*u_z*u_xy*v_y*w_yz - 24*u_y*u_xz*v_y*w_yz + 24*u_xz*v_x*v_y*w_yz + 48*u_y*u_xy*v_z*w_yz - 24*u_xx*v_y*v_z*w_yz - 24*u_x*v_z*v_xy*w_yz + 24*u_x*u_z*v_yy*w_yz + 24*u_x*u_y*v_yz*w_yz - 24*u_x*v_x*v_yz*w_yz - 24*u_xy*v_y*w_x*w_yz + 48*v_x*v_xy*w_x*w_yz - 24*u_x*v_yy*w_x*w_yz + 48*u_x*u_yy*w_y*w_yz - 48*u_xy*v_x*w_y*w_yz + 24*u_xx*v_y*w_y*w_yz - 24*u_x*v_xy*w_y*w_yz + 24*u_y^2*u_yy*w_zz - 24*u_y*u_xy*v_y*w_zz - 24*u_x*u_yy*v_y*w_zz + 24*u_xy*v_x*v_y*w_zz - 24*v_x^2*v_xy*w_zz + 24*u_x*v_y*v_xy*w_zz - 24*u_x*u_y*v_yy*w_zz + 24*u_x*v_x*v_yy*w_zz)*xi1*xi3 + (8*u_z^3*u_yyy - 24*u_y*u_z^2*u_yyz + 24*u_y^2*u_z*u_yzz - 8*u_y^3*u_zzz + 8*u_zzz*v_x^3 + 24*u_z^2*u_xyz*v_y - 24*u_y*u_z*u_xzz*v_y - 24*u_x*u_z*u_yzz*v_y + 24*u_x*u_y*u_zzz*v_y + 24*u_z*u_xzz*v_x*v_y - 24*u_x*u_zzz*v_x*v_y - 24*u_z*u_xz*u_yy*v_z + 24*u_z*u_xy*u_yz*v_z + 24*u_y*u_xz*u_yz*v_z - 24*u_x*u_yz^2*v_z - 24*u_y*u_xy*u_zz*v_z + 24*u_x*u_yy*u_zz*v_z - 24*u_z^2*u_xyy*v_z + 24*u_y*u_z*u_xyz*v_z + 24*u_x*u_z*u_yyz*v_z - 24*u_x*u_y*u_yzz*v_z + 24*u_xz*u_yz*v_x*v_z - 24*u_xy*u_zz*v_x*v_z - 24*u_z*u_xyz*v_x*v_z + 24*u_x*u_yzz*v_x*v_z - 24*u_xzz*v_x^2*v_z - 24*u_xz^2*v_y*v_z + 24*u_xx*u_zz*v_y*v_z - 24*u_z*u_xxz*v_y*v_z + 24*u_x*u_xzz*v_y*v_z + 24*u_xy*u_xz*v_z^2 - 24*u_xx*u_yz*v_z^2 + 24*u_z*u_xxy*v_z^2 - 24*u_x*u_xyz*v_z^2 + 24*u_xxz*v_x*v_z^2 - 8*u_xxx*v_z^3 + 24*u_xz*v_z^2*v_xx - 48*u_z*u_xz*v_z*v_xy - 24*u_zz*v_x^2*v_xz + 24*u_z*u_xz*v_y*v_xz + 24*u_x*u_zz*v_y*v_xz + 24*u_z*u_xy*v_z*v_xz - 24*u_x*u_yz*v_z*v_xz - 24*u_xx*v_z^2*v_xz + 24*u_z^2*u_xz*v_yy - 24*u_z^2*u_xy*v_yz - 24*u_y*u_z*u_xz*v_yz + 24*u_x*u_z*u_yz*v_yz - 24*u_x*u_y*u_zz*v_yz - 24*u_z*u_xz*v_x*v_yz + 24*u_x*u_zz*v_x*v_yz + 24*u_z*u_xx*v_z*v_yz + 24*u_x*u_xz*v_z*v_yz + 24*u_y*u_z*u_xy*v_zz - 24*u_x*u_z*u_yy*v_zz + 24*u_x*u_y*u_yz*v_zz + 24*u_z*u_xy*v_x*v_zz - 24*u_x*u_yz*v_x*v_zz + 24*u_xz*v_x^2*v_zz - 24*u_z*u_xx*v_y*v_zz - 24*u_x*u_xz*v_y*v_zz - 24*u_yzz*v_x^2*w_x - 24*u_y*u_xzz*v_y*w_x + 24*u_x*u_yzz*v_y*w_x + 24*u_xzz*v_x*v_y*w_x - 24*u_xz*u_yy*v_z*w_x + 24*u_xy*u_yz*v_z*w_x + 24*u_y*u_xyz*v_z*w_x - 24*u_x*u_yyz*v_z*w_x + 24*u_xyz*v_x*v_z*w_x - 24*u_xxz*v_y*v_z*w_x + 48*u_yz*v_x*v_xz*w_x - 24*u_xz*v_y*v_xz*w_x + 24*u_xy*v_z*v_xz*w_x + 24*u_y*u_xz*v_yz*w_x - 24*u_x*u_yz*v_yz*w_x - 24*u_xz*v_x*v_yz*w_x - 24*u_xx*v_z*v_yz*w_x - 24*u_y*u_xy*v_zz*w_x + 24*u_x*u_yy*v_zz*w_x - 24*u_xy*v_x*v_zz*w_x + 24*u_xx*v_y*v_zz*w_x + 24*u_yyz*v_x*w_x^2 - 24*u_xyz*v_y*w_x^2 - 24*u_yy*v_xz*w_x^2 + 24*u_xy*v_yz*w_x^2 - 8*u_yyy*w_x^3 + 24*u_z*u_xz*u_yy*w_y - 24*u_z*u_xy*u_yz*w_y - 24*u_y*u_xz*u_yz*w_y + 24*u_x*u_yz^2*w_y + 24*u_y*u_xy*u_zz*w_y - 24*u_x*u_yy*u_zz*w_y - 24*u_y*u_z*u_xyz*w_y + 24*u_y^2*u_xzz*w_y + 24*u_x*u_z*u_yyz*w_y - 24*u_x*u_y*u_yzz*w_y - 24*u_xz*u_yz*v_x*w_y + 24*u_xy*u_zz*v_x*w_y - 24*u_z*u_xyz*v_x*w_y + 24*u_x*u_yzz*v_x*w_y + 24*u_xz^2*v_y*w_y - 24*u_xx*u_zz*v_y*w_y + 24*u_z*u_xxz*v_y*w_y - 24*u_x*u_xzz*v_y*w_y - 24*u_z*u_xy*v_xz*w_y - 24*u_x*u_yz*v_xz*w_y + 24*u_z*u_xx*v_yz*w_y + 24*u_x*u_xz*v_yz*w_y + 24*u_xz*u_yy*w_x*w_y - 24*u_xy*u_yz*w_x*w_y + 24*u_y*u_xyz*w_x*w_y - 24*u_x*u_yyz*w_x*w_y - 24*u_xyz*v_x*w_x*w_y + 24*u_xxz*v_y*w_x*w_y + 24*u_xy*v_xz*w_x*w_y - 24*u_xx*v_yz*w_x*w_y + 24*u_xyy*w_x^2*w_y - 24*u_xy*u_xz*w_y^2 + 24*u_xx*u_yz*w_y^2 - 24*u_y*u_xxz*w_y^2 + 24*u_x*u_xyz*w_y^2 - 24*u_xxy*w_x*w_y^2 + 8*u_xxx*w_y^3 + 24*u_y*u_z*u_xyy*w_z - 24*u_y^2*u_xyz*w_z - 24*u_x*u_z*u_yyy*w_z + 24*u_x*u_y*u_yyz*w_z + 24*u_z*u_xyy*v_x*w_z - 24*u_x*u_yyz*v_x*w_z + 24*u_xyz*v_x^2*w_z - 24*u_z*u_xxy*v_y*w_z + 24*u_y*u_xxz*v_y*w_z - 24*u_xxz*v_x*v_y*w_z - 24*u_xy^2*v_z*w_z + 24*u_xx*u_yy*v_z*w_z - 24*u_y*u_xxy*v_z*w_z + 24*u_x*u_xyy*v_z*w_z - 24*u_xxy*v_x*v_z*w_z + 24*u_xxx*v_y*v_z*w_z - 24*u_xz*v_y*v_xx*w_z - 24*u_xy*v_z*v_xx*w_z + 24*u_z*u_xy*v_xy*w_z + 24*u_y*u_xz*v_xy*w_z + 24*u_xz*v_x*v_xy*w_z + 24*u_xx*v_z*v_xy*w_z - 24*u_y*u_xy*v_xz*w_z + 24*u_x*u_yy*v_xz*w_z - 24*u_xy*v_x*v_xz*w_z + 24*u_xx*v_y*v_xz*w_z - 24*u_z*u_xx*v_yy*w_z - 24*u_x*u_xz*v_yy*w_z - 24*u_y*u_xyy*w_x*w_z + 24*u_x*u_yyy*w_x*w_z - 24*u_xyy*v_x*w_x*w_z + 24*u_xxy*v_y*w_x*w_z - 24*u_xy*v_xy*w_x*w_z + 24*u_xx*v_yy*w_x*w_z + 24*u_xy^2*w_y*w_z - 24*u_xx*u_yy*w_y*w_z + 24*u_y*u_xxy*w_y*w_z - 24*u_x*u_xyy*w_y*w_z + 24*u_xxy*v_x*w_y*w_z - 24*u_xxx*v_y*w_y*w_z + 24*u_xy*v_xx*w_y*w_z - 24*u_xx*v_xy*w_y*w_z - 24*u_xz*v_y*v_z*w_xx + 48*v_x*v_z*v_xz*w_xx - 24*u_x*v_z*v_yz*w_xx - 24*v_x^2*v_zz*w_xx + 24*u_x*v_y*v_zz*w_xx - 48*v_y*v_xz*w_x*w_xx + 48*v_x*v_yz*w_x*w_xx - 24*v_yy*w_x^2*w_xx + 24*u_xz*v_y*w_y*w_xx - 24*u_x*v_yz*w_y*w_xx + 48*v_xy*w_x*w_y*w_xx - 24*u_xy*w_y^2*w_xx + 24*u_xy*v_y*w_z*w_xx - 48*v_x*v_xy*w_z*w_xx + 24*u_x*v_yy*w_z*w_xx + 24*u_zz*v_x^2*w_xy + 24*u_z*u_xz*v_y*w_xy - 24*u_x*u_zz*v_y*w_xy + 24*u_y*u_xz*v_z*w_xy + 24*u_x*u_yz*v_z*w_xy - 24*u_xz*v_x*v_z*w_xy - 48*u_z*v_x*v_xz*w_xy - 24*u_x*v_z*v_xz*w_xy + 24*u_x*u_z*v_yz*w_xy - 24*u_x*u_y*v_zz*w_xy + 24*u_x*v_x*v_zz*w_xy - 48*u_yz*v_x*w_x*w_xy + 24*u_xz*v_y*w_x*w_xy + 48*u_y*v_xz*w_x*w_xy - 24*u_x*v_yz*w_x*w_xy + 24*u_yy*w_x^2*w_xy - 24*u_y*u_xz*w_y*w_xy + 24*u_x*u_yz*w_y*w_xy - 24*u_xz*v_x*w_y*w_xy + 24*u_x*v_xz*w_y*w_xy - 48*v_xx*w_x*w_y*w_xy + 24*u_xx*w_y^2*w_xy - 24*u_y*u_xy*w_z*w_xy - 24*u_x*u_yy*w_z*w_xy + 24*u_xy*v_x*w_z*w_xy - 24*u_xx*v_y*w_z*w_xy + 48*v_x*v_xx*w_z*w_xy - 24*u_z*u_xy*v_y*w_xz - 24*u_y*u_xz*v_y*w_xz + 24*u_xz*v_x*v_y*w_xz + 24*u_xx*v_y*v_z*w_xz - 48*v_x*v_z*v_xx*w_xz + 48*u_z*v_x*v_xy*w_xz + 24*u_x*v_z*v_xy*w_xz - 24*u_x*u_z*v_yy*w_xz + 24*u_x*u_y*v_yz*w_xz - 24*u_x*v_x*v_yz*w_xz - 24*u_xy*v_y*w_x*w_xz + 48*v_y*v_xx*w_x*w_xz - 48*u_y*v_xy*w_x*w_xz + 24*u_x*v_yy*w_x*w_xz + 48*u_y*u_xy*w_y*w_xz - 24*u_xx*v_y*w_y*w_xz - 24*u_x*v_xy*w_y*w_xz - 24*u_y*u_z*u_xz*w_yy - 24*u_x*u_z*u_yz*w_yy + 24*u_x*u_y*u_zz*w_yy + 24*u_z*u_xz*v_x*w_yy - 24*u_x*u_zz*v_x*w_yy + 24*u_x*u_z*v_xz*w_yy - 24*u_y*u_xz*w_x*w_yy + 24*u_x*u_yz*w_x*w_yy + 24*u_xz*v_x*w_x*w_yy - 24*u_x*v_xz*w_x*w_yy - 24*u_xy*w_x^2*w_yy + 24*v_xx*w_x^2*w_yy + 24*u_y*u_xx*w_z*w_yy + 24*u_x*u_xy*w_z*w_yy - 24*u_xx*v_x*w_z*w_yy - 24*u_x*v_xx*w_z*w_yy + 24*u_y*u_z*u_xy*w_yz + 24*u_y^2*u_xz*w_yz + 24*u_x*u_z*u_yy*w_yz - 24*u_x*u_y*u_yz*w_yz - 24*u_z*u_xy*v_x*w_yz + 24*u_x*u_yz*v_x*w_yz - 24*u_xz*v_x^2*w_yz - 24*u_y*u_xx*v_z*w_yz - 24*u_x*u_xy*v_z*w_yz + 24*u_xx*v_x*v_z*w_yz + 24*u_x*v_z*v_xx*w_yz - 24*u_x*u_z*v_xy*w_yz - 24*u_x*u_y*v_xz*w_yz + 24*u_x*v_x*v_xz*w_yz + 24*u_y*u_xy*w_x*w_yz - 24*u_x*u_yy*w_x*w_yz + 24*u_xy*v_x*w_x*w_yz - 48*v_x*v_xx*w_x*w_yz + 24*u_x*v_xy*w_x*w_yz - 24*u_y*u_xx*w_y*w_yz - 24*u_x*u_xy*w_y*w_yz + 24*u_xx*v_x*w_y*w_yz + 24*u_x*v_xx*w_y*w_yz - 24*u_y^2*u_xy*w_zz + 24*u_y*u_xx*v_y*w_zz + 24*u_x*u_xy*v_y*w_zz - 24*u_xx*v_x*v_y*w_zz + 24*v_x^2*v_xx*w_zz - 24*u_x*v_y*v_xx*w_zz + 24*u_x*u_y*v_xy*w_zz - 24*u_x*v_x*v_xy*w_zz)*xi2*xi3

Is a Poisson 2-coboundary: ?

Code (gcaops):

from gcaops.all import *

D = DifferentialPolynomialRing(QQ, fibre_names=['u','v','w'], base_names=['x','y','z'], max_differential_orders=[4,4,4])
u,v,w = D.fibre_variables()
x,y,z = D.base_variables()

S.<xi1,xi2,xi3> = SuperfunctionAlgebra(D, (x, y, z))
P = S(u)*xi2*xi3 + S(v)*xi3*xi1 + S(w)*xi1*xi2

GC = UndirectedGraphComplex(QQ, implementation='vector', sparse=True)
gamma3 = GC.cohomology_basis(4,6)[0]
gamma3_op_3d = S.graph_operation(gamma3)

Q_gamma3_3d = S.zero()
try:
    Q_gamma3_3d = load('Q_gamma3_3d.sobj')
except FileNotFoundError:
    Q_gamma3_3d = gamma3_op_3d(P,P,P,P)
    save(Q_gamma3_3d, 'Q_gamma3_3d.sobj')
    with open('Q_gamma3_3d.txt', 'w') as f:
        print(Q_gamma3_3d, file=f)

try:
    L_gamma3_3d = load('L_gamma3_3d.sobj')
except FileNotFoundError:
    L_gamma3_3d = gamma3_op_3d(P,P,P,P.bracket(P))
    P_Q_gamma3_3d = P.bracket(Q_gamma3_3d)
    assert P_Q_gamma3_3d == 2*L_gamma3_3d
    save(L_gamma3_3d, 'L_gamma3_3d.sobj')
    with open('L_gamma3_3d.txt', 'w') as f:
        print(L_gamma3_3d, file=f)