For any category of interest C we define a general category of groups with operations C G , C → C G , and a universal strict general actor USGA(A) of an object A in C, which is an object of C G . The notion of actor is equivalent to the one of split extension classifier defined for an object in more general settings of semi-abelian categories. It is proved that there exists an actor of A in C if and only if the semidirect product USGA(A) A is an object of C and if it is the case, then USGA(A) is an actor of A. We give a construction of a universal strict general actor for any A ∈ C, which helps to detect more properties of this object. The cases of groups, Lie, Leibniz, associative, commutative associative, alternative algebras, crossed and precrossed modules are considered. The examples of algebras are given, for which always exist actors.
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.
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