Automorphisms of the Category of the Free Nilpotent Groups of the Fixed Class of Nilpotency

This research was motivated by universal algebraic geometry. One of the central questions of universal algebraic geometry is: when two algebras have the same algebraic geometry? For answer of this question (see [8],[10]) we must consider the variety Θ, to which our algebras belongs, the category Θ of all finitely generated free algebras of Θ and research how the group AutΘ of all the automorphisms of the category Θ are different from the group InnΘ of the all inner automorphisms of the category Θ. An automorphism Υ of the arbitrary category K is called inner, if it is isomorphic as functor to the identity automorphism of the category K, or, in details, for every A ∈ ObK there exists sΥA : A → Υ(A) isomorphism of these objects of the category K and for every α ∈ MorK (A,B) the diagram A −→ sΥA Υ(A) ↓ α Υ(α) ↓