The notions of left-right noncommutative Poisson algebra (NPlr-algebra) and left-right algebra with bracket AWBlr are introduced. These algebras are special cases of NLP-algebras and algebras with bracket AWB, respectively, studied earlier. An NPlr-algebra is a noncommutative analogue of the classical Poisson algebra. Properties of these new algebras are studied. In the categories AWBlr and NPlr-algebras the notions of actions, representations, centers, actors and crossed modules are described as...

A finite-dimensional Lie algebra is called an $A$-algebra if all of its nilpotent subalgebras are abelian. These arise in the study of constant Yang-Mills potentials and have also been particularly important in relation to the problem of describing residually finite varieties. They have been studied by several authors, including Bakhturin, Dallmer, Drensky, Sheina, Premet, Semenov, Towers and Varea. In this paper we establish generalisations of many of these results to Leibniz algebras.

We propose a definition of Leibniz cohomology, $H{L}^{*}$, for differentiable manifolds. Then $H{L}^{*}$ becomes a non-commutative version of Gelfand-Fuks cohomology. The calculations of $H{L}^{*}({\mathbf{R}}^n;\mathbf{R})$ reduce to those of formal vector fields, and can be identified with certain invariants of foliations.