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 the new algebras are studied. The constructions of free objects in the corresponding categories are given. The relations between the properties of $\NP^{lr}$-algebras, the underlying $\AWB^{lr}$, associative and Leibniz algebras are investigated. 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 well-known 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 $\NP^{r}$ or $\NP^{l}$-algebra, the Hochschild or/and Leibniz cohomological dimension of $P$ is $\leq n$ are considered separately, exhibiting interesting possibilities of representations of the new cohomologies by the well-known ones and relations between the corresponding cohomological dimensions.