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

JOURNAL OF ALGEBRA

卷:426

Article

Agore, A. L.^1,3  Militaru, G.² 

[1] Vrije Univ Brussel, Fac Engn, B-1050 Brussels, Belgium

[2] Univ Bucharest, Fac Math & Comp Sci, RO-010014 Bucharest 1, Romania

[3] Bucharest Univ Econ Studies, Dept Appl Math, RO-010374 Bucharest 1, Romania

关键词: The extension problem for Poisson;    algebras;    Crossed products;    Classification results;   

DOI  :  10.1016/j.jalgebra.2014.12.007

来源: Elsevier

【 摘 要 】

Let P be a Poisson algebra, E a vector space and pi : E -> P an epimorphism of vector spaces with V = Ker(pi). The global extension problem asks for the classification of all Poisson algebra structures that can be defined on E such that pi : 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 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 H-2(P, V) appears as the most elementary piece among all components of gPH(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 finite chain of epimorphisms of Poisson algebras P-n := Q (pi n)-> Pn-1 . . . P-1 pi(1) -> P-0 := P such that dim(Ker(pi(i))) = 1, for all i = 1, . . . , n. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free