Examples of Poisson modules, I

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

“…for all p, q ∈ P , x, y ∈ V . P ⋆ V is called the Hochschild (or crossed) product associated to Θ(P, V ) if it is an associative algebra with the multiplication given by (9). In this case the Hochschild datum Θ(P, V ) = (⇀, ⊳, ϑ, ·) is called a Hochschild (or crossed) system of P by V and we denote by HS(P, V ) the set of all Hochschild systems of P by V .…”

“…Beyond the remarkable applications in the above mentioned fields, Poisson algebras are objects of study in their own right, from a purely algebraic viewpoint [7,8,9,10,14,16,21,22,28,29,35] etc. In this spirit a tempting question arises: for a given positive integer n, classify up to an isomorphism all Poisson algebras of dimension n over a field k. Having in mind the duality established by the functor C ∞ (−), the question can be viewed as the algebraic version of its geometric counterpart initiated in [17] where the first steps towards the classification of low dimensional Poisson structures is given using differential geometry tools.…”

“…Lie, Poisson, Hopf, etc.) algebras of a given dimension, relies on the famous extension problem initiated at the level of groups by Hölder and studied later on for Lie algebras [12], associative algebras [20], quantum groups [5], Poisson algebras [9], etc. For the relevance of the extension problem in differential geometry we refer to [4,26].…”

“…We start by explaining the meaning of the word 'global'. We recall the classical extension problem formulated in the setting of Poisson algebras in [9]. Let P and Q be two fixed Poisson algebras.…”

“…Moreover, we shall prove that GPH 2 (P, V ) is the coproduct in the category of sets of all classifying objects of all 'local' extension problems, which are the non-abelian cohomological objects denoted by PH 2 (P, (V, · V , [−, −] V )) -for any Poisson algebra structure (· V , [−, −] V ) on V . The second classical Poisson cohomology group H 2 (P, V ) [9,15] appears as the most elementary piece among all components of GPH 2 (P, V ). The results are proved in the general case, leaving aside the 'abelian' case which is the traditional setting for the extension problem for groups or Lie (resp.…”