# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Poisson (co)homology and isolated singularities. (English) Zbl 1113.17009
Let $𝔽$ be a field of characteristic 0 and $𝒜=𝔽\left[x,y,z\right]$. Given any $\varphi \in 𝒜$, the relations ${\left\{x,y\right\}}_{\varphi }=\frac{\partial \varphi }{\partial z}$, ${\left\{y,z\right\}}_{\varphi }=\frac{\partial \varphi }{\partial x}$, ${\left\{z,x\right\}}_{\varphi }=\frac{\partial \varphi }{\partial y}$ define a Poisson bracket on $𝒜$, which admits $\varphi$ as a Casimir function. Therefore, this bracket induces Poisson structures both on the affine three space ${F}^{3}$ and the surface $\left\{\varphi =0\right\}\subset {F}^{3}$. Suppose that $\varphi$ is a weighted homogeneous polynomial such that the surface $\left\{\varphi =0\right\}$ has an isolated singularity at the origin. The author computes the Poisson cohomology and homology modules of the Poisson structures on ${F}^{3}$ and $\left\{\varphi =0\right\}$ in this case. The paper also contains clear explanations of each of the concepts mentioned.
##### MSC:
 17B63 Poisson algebras 14F99 Homology and cohomology theory (algebraic geometry) 17B56 Cohomology of Lie (super)algebras