Automorphisms of the semigroup of endomorphisms of free algebras of homogeneous varieties

TL;DR

This paper investigates the structure of automorphisms of the semigroup of endomorphisms of free algebras in homogeneous varieties over R1MF-domains, establishing semi-innerity results and describing automorphism groups.

Contribution

It proves semi-innerity of automorphisms for free finitely generated Lie algebras over R1MF-domains and describes automorphism groups for various algebraic varieties, extending previous results.

Findings

Automorphisms of EndF are semi-inner for free finitely generated Lie algebras over R1MF-domains.

The group AutEndF is explicitly described for m-nilpotent associative algebras.

Not all automorphisms are quasi-inner in some varieties over Dedekind domains.

Abstract

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 be a free algebra of some variety A of linear algebras over K freely generated by a finite set X, EndF be the semigroup of endomorphisms of F, and AutEndF be the group of automorphisms of the semigroup EndF. We investigate structure of the group AutEndF and its relation to the algebraical and categorical equivalence of algebras from A. 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 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 R1MF-domains are clarified. The group AutEndF 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 x n matrices over R1MF-domains is obtained. We give an example of the variety of linear algebras over a Dedekind domain such that not all automorphisms of AutEndF are quasi-inner. The results obtained generalize the previous studies of various special cases of varieties of linear algebras over infinite fields.