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

Abstract

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 GPH 2 (P, V ) which is the coproduct of all non-abelian cohomological objects PH

On Nowicki's conjecture: a survey and a new result

“…] = 0 holds in P , then the Poisson algebra P is called abelian and any extension of P by an abelian Poisson algebra is called metabelian (see e.g. [1]).…”

Hochschild products and global non-abelian cohomology for algebras. Applications

“…To start with, we can easily prove that (1 A , 0 V ) is the unit for the multiplication defined by (3) if and only if (H0) holds. The rest of the proof relies on a detailed analysis of the associativity condition for the multiplication given by (3). Since in A ⋆ V we have (a, x) = (a, 0) + (0, x), it follows that the associativity condition holds if and only if it holds for all generators of A⋆V , i.e.…”

“…It what follows we shall classify all Poisson algebras Q that admit a surjective Poisson algebra map Q → P → 0 with a 1-dimensional kernel. In order to do this we first recall from [3] the following concept: (2) Two Poisson algebras P (λ, Λ, ϑ, γ, f ) and P (λ ′ , Λ ′ , ϑ ′ , γ ′ , f ′ ) are isomorphic if and only if there exists a triple (s 0 , ψ, r) ∈ k * ×Aut Poss (P )×Hom k (P, k) such that for any p, q ∈ P : which is precisely (19). The first part is finished once we observe that the map given by the formula (21), namely ϕ : P (λ, u) → P × k, ϕ(p, x) := (p, λ(p) + u x), for all p ∈ P and x ∈ k is an isomorphism of Poisson algebras.…”

Extending structures for weighted infinitesimal bialgebras

“…The theory of extending structure for many types of algebras were well developed by A. L. Agore and G. Militaru in [1,2,3,4,5,6]. Recently, we developed the theory of extending structure for infinitesimal bialgebras in [31].…”