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Poisson (co)homology and isolated singularities. (English) Zbl 1113.17009
Let 𝔽 be a field of characteristic 0 and 𝒜=𝔽[x,y,z]. Given any ϕ𝒜, the relations {x,y} ϕ =ϕ z, {y,z} ϕ =ϕ x, {z,x} ϕ =ϕ y define a Poisson bracket on 𝒜, which admits ϕ as a Casimir function. Therefore, this bracket induces Poisson structures both on the affine three space F 3 and the surface {ϕ=0}F 3 . Suppose that ϕ is a weighted homogeneous polynomial such that the surface {ϕ=0} has an isolated singularity at the origin. The author computes the Poisson cohomology and homology modules of the Poisson structures on F 3 and {ϕ=0} in this case. The paper also contains clear explanations of each of the concepts mentioned.
MSC:
17B63Poisson algebras
14F99Homology and cohomology theory (algebraic geometry)
17B56Cohomology of Lie (super)algebras