  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.
