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...

On étudie dans cet article les notions d’algèbre à homotopie près pour une structure définie par deux opérations $\text{.}$ et $[,]$. Ayant déterminé la structure des $G_{\infty }$ algèbres et des $P_{\infty }$ algèbres, on généralise cette construction et on définit la stucture des $(a,b)$-algèbres à homotopie près. Etant donnée une structure d’algèbre commutative et de Lie différentielle graduée pour deux décalages des degrés donnés par $a$ et $b$, on donnera une construction explicite de l’algèbre à homotopie près associée et on précisera...

Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class of diffeomorphisms of the underlying manifolds.

For any Lie-Rinehart algebra $(A,L)$, B(atalin)-V(ilkovisky) algebra structures $\partial$ on the exterior $A$-algebra ${\Lambda }_{A}L$ correspond bijectively to right $(A,L)$-module structures on $A$; likewise, generators for the Gerstenhaber algebra ${\Lambda }_{A}L$ correspond bijectively to right $(A,L)$-connections on $A$. When $L$ is projective as an $A$-module, given a B-V algebra structure $\partial$ on ${\Lambda }_{A}L$, the homology of the B-V algebra $({\Lambda }_{A}L,\partial )$ coincides with the homology of $L$ with coefficients in $A$ with reference to the right $(A,L)$-module structure determined by $\partial$. When...