Left-right noncommutative Poisson algebras

Abstract

The notions of left-right noncommutative Poisson algebra (NP lr -algebra) and left-right algebra with bracket AWB lr are introduced. These algebras are special cases of NLP-algebras and algebras with bracket AWB, respectively, studied earlier. An NP lr -algebra is a noncommutative analogue of the classical Poisson algebra. Properties of these new algebras are studied. In the categories AWB lr and NP lr -algebras the notions of actions, representations, centers, actors and crossed modules are described as special cases of the corresponding wellknown notions in categories of groups with operations. The cohomologies of NP lr -algebras and AWB lr (resp. of NP r -algebras and AWB r ) are defined and the relations between them and the Hochschild, Quillen and Leibniz cohomologies are detected. The cases P is a free AWB r , the Hochschild or/and Leibniz cohomological dimension of P is ≤ are considered separately, exhibiting interesting possibilities of representations of the new cohomologies by the well-known ones and relations between the corresponding cohomological dimensions.

From simplicial homotopy to crossed module homotopy in modified categories of interest

“…The main idea is due to Higgins [13] and the definition is improved by Orzech [19]. As indicated in [9,10,15,16,17,19], many algebraic categories are the essential examples of category of interest. However the categories of cat 1 -objects of Lie (associative, Leibniz, etc.)…”

The structures on the universal enveloping algebras of differential graded Poisson Hopf algebras

“…Poisson algebras appear naturally in Hamiltonian mechanics, and play a central role in the study of Poisson geometry and quantum groups. With the development of Poisson algebras in the past decades, many important generalizations have been obtained in both commutative and noncommutative settings: Poisson PI algebras [10], graded Poisson algebras [3], double Poisson algebras [20], Quiver Poisson algebras [22], noncommutative Leibniz-Poisson algebras [1], Left-right noncommutative Poisson algebras [2] and differential graded Poisson algebras [8], etc. One of most interesting features in this area is the Poisson universal enveloping algebra, which was first introduced by Oh [12] in order to describe the category of Poisson modules.…”

“…Note that the universal enveloping algebra A e of a Poisson bialgebra A is a bialgebra, as an extension, we prove the following result: It should be noted that the universal enveloping algebra A e of a DG Poisson Hopf algebra A is just a DG bialgebra. But in some special cases, A e can be endowed with the Hopf structure, such that A e becomes a DG Hopf algebra: (1) ), a (2) } = 0, where ∆(a) = a (1) ⊗ a (2) for all a ∈ A, if and only if (A e , u e , η e , ∆ e , ε e , S e , d e ) is a DG Hopf algebra.…”

On Solvability and Nilpotency of Algebras With Bracket

“…It is enough to use the following obvious equivalence: (A). Since AWB is a category of Ω-groups, but not a category of interest (see [8]), and following Proposition 4.3.2 in [14] we can conclude that the collection of all nilpotent objects of class ≤ k in AWB form a variety. Now following [1,2,3,11,17] can be showed that M (c) (A) is a Baer-invariant, which means that its definition does not depend on the choice of the free presentation.…”

“…Since the seminal paper [9], several algebraic properties of this structure, in particular the homological ones, were analyzed in different articles [6,7,8].…”