Automorphisms of the Semigroup of Endomorphisms of Free Algebras of Homogeneous Varieties

We consider homogeneous varieties of linear algebras over an associative-commutative ring K with 1, i.e., the varieties in which free algebras are graded. Let F = F (x1, ..., xn) be a free algebra of some variety Θ of linear algebras over K freely generated by a set X = {x1, ..., xn}, EndF be the semigroup of endomorphisms of F , and Aut EndF be the group of automorphisms of the semigroup EndF . We investigate structure of the group Aut End F and its relation to the algebraical and categorical equivalence of algebras from Θ. We define a wide class of R1MF-domains containing, in particular, Bezout domains, unique factorization domains, and some other domains. We show that every automorphism Φ of semigroup EndF , where F is a free finitely generated Lie algebra over an R1MF-domain, is semi-inner. This solves the Problem 5.1 left open in [21]. As a corollary, semi-innerity of all automorphism of the category of free Lie algebras over R1MF-domains is obtained. Relations between categorical and geometrical equivalence of Lie algebras over R1MFdomains are clarified. The group Aut EndF for the variety of m-nilpotent associative algebras over R1MF-domains is described. As a consequence, a complete description of the group of automorphisms of the full matrix semigroup of n × n matrices over R1MF-domains is obtained. We give an example of the variety Θ of linear algebras over a Dedekind domain such that not all automorphisms of Aut EndF are quasi-inner. The results obtained generalize the previous studies of various special cases of varieties of linear algebras over infinite fields.