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

Abstract

Let P be a Poisson algebra, E a vector space and π:E→P an epimorphism of vector spaces with V=Ker(π). The global extension problem asks for the classification of all Poisson algebra structures that can be defined on E such that π:E→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 GPH2(P,V) which is the coproduct of all non-abelian cohomological objects PH2(P,(V,⋅V,[−,−]V)) which are the classifying sets for all extensions of P by (V,⋅V,[−,−]V). The second classical Poisson cohomology group H2(P,V) appears as the most elementary piece among all components of GPH2(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 Pn:=Q⟶πnPn−1⋯P1⟶π1P0:=P such that dim⁡(Ker(πi))=1, for all i=1,⋯,n.