Show code: [gcaops]

The sea of Poisson Structures and its Graph Complex flows

Index » Generic Poisson structure on $\mathbb{R}^3$

Alternative names:
  • Arbitrary Poisson structure on $\mathbb{R}^3$
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Definition

Formula:$P = u \frac{\partial}{\partial y} \wedge \frac{\partial}{\partial z} + v \frac{\partial}{\partial z} \wedge \frac{\partial}{\partial x} + w \frac{\partial}{\partial x} \wedge \frac{\partial}{\partial y}$ with $u,v,w \in C^\infty(\mathbb{R}^3)$ such that $[P,P]=0$.
Code (gcaops):
D = DifferentialPolynomialRing(QQ, fibre_names=['u','v','w'], base_names=['x','y','z'], max_differential_orders=[1,1,1])
x,y,z = D.base_variables()
u,v,w = D.fibre_variables()

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

Deformations