Automorfismos da categoria de álgebras livres associativas e comutativas com unidade

Abstract

In Universal Algebraic Geometry over a variety of algebras, the group of automorphisms of the category of free finitely generated algebras plays an important role. It turns out that for the wide class of varieties, an automorphism of the category of free finitely generated algebras can be decomposed to the product of an inner automorphism and a strongly stable automorphism. The method of verbal operations provides a machinery to calculate the group of strongly stable automorphisms. In the present work, we deal with the variety of associative and commutative algebras with unit over a field of zero characteristic. Note that free finitely generated algebras in this variety are exactly algebras of commutative and associative polynomials with unit over a finite set of variables. The main goal of this dissertation is to investigate automorphisms of the category of free finitely generated algebras in this variety. In particular, applying the method of verbal operations we get description of group of the strongly stable automorphisms of the category under consideration.