The global extension problem, crossed products and co-flag non-commutative Poisson algebras

Let $P$ be a Poisson algebra, $E$ a vector space and $\pi :E\to P$ an epimorphism of vector spaces with $V=\mathrm{K}\mathrm{e}\mathrm{r}(\pi )$. The global extension problem asks for the classification of all Poisson algebra structures that can be defined on $E$ such that $\pi :E\to P$ becomes a morphism of Poisson algebras. From a geometrical point of view it means to decompose this groupoid into connected components and to indicate a point in each such component. All such Poisson algebra structures on $E$ are classified by an explicitly constructed classifying set ${\mathcal G} {\mathcal P} {\mathcal H}^{2} , (P, , V)$whichisthecoproductofallnon-abeliancohomologicalobjects${\mathcal P} {\mathcal H}^{2} , (P, , (V, \cdot_V, [-,-]V))$ which are the classifying sets for all extensions of $P$ by $(V,{\cdot }_V,[-,-{]}_V)$. The second classical Poisson cohomology group $H^2(P,V)$ appears as the most elementary piece among all components of ${\mathcal G} {\mathcal P} {\mathcal H}^{2} , (P, , V)$. Several examples are provided in the case of metabelian Poisson algebras or co-flag Poisson algebras over $P$: the latter being Poisson algebras $Q$ which admit a finite chain of epimorphisms of Poisson algebras $P_n : = Q \stackrel{\pi{n}}{\longrightarrow} P_{n-1} , \cdots , P_1 \stackrel{\pi_{1}} {\longrightarrow} P_{0} := P$suchthat${\rm dim} ( {\rm Ker} (\pi_{i}) ) = 1$,forall$i = 1, \cdots, n$.