Aupomorphisms of the category of the free nilpotent groups of the fixed class of nilpotency

Abstract

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 [Pl],[Ts]) we must consider the variety, to which our algebras belongs, the category K of all finitely generated free algebras of our variety and research how the group of all the automorphisms of this category AutK are different from the group of the all inner automorphisms of this category InnK. An automorphism of the category is called inner, if it is isomorphic as functor to the identity automorphism. In the case when we consider a variety of all groups we have the classical results which let us resolve this problem by an indirect way. In [DF] proved that for every free group F the group Aut(AutF) coincides with the group Inn(AutF), from this result in [Fo] was concluded that Aut(EndF)=Inn(EndF) and from this fact by theorem of reduction [BPP] it can be concluded that AutK=InnK. In the case of nilpotent group by [Ks] we have, that if the number of generators of the free nilpotent class d group NF are bigger enough than d, then Aut(AutNF)=Inn(AutNF). But we have no description of Aut(EndNF) and so can not use the theorem of reduction. In this paper the method of verbal operations is used. This method was established in [PZ]. In [PZ] by this method was very easily proved that AutK=InnK in the case of free groups and in the case of free abelian groups. In this paper we will prove, by this method, that AutK=InnK free nilpotent groups of arbitrary fixed class of nilpotency.