Automorphisms of the endomorphism semigroups of free linear algebras of homogeneous varieties

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 = F ( x 1 , … , x n ) be a free algebra of some variety Θ of linear algebras over K freely generated by a set X = { x 1 , … , x n } , End F be the semigroup of endomorphisms of F, and Aut End F be the group of automorphisms of the semigroup End F. We investigate the structure of the group Aut End F and its relation to the algebraic and categorical equivalence of algebras from Θ. We define a wide class of R 1 MF -domains containing, in particular, Bezout domains, unique factorization domains, and some other domains. We show that every automorphism Φ of semigroup End F, where F is a free finitely generated Lie algebra over an R 1 MF -domain, is semi-inner. This solves the Problem 5.1 left open in [G. Mashevitzky, B. Plotkin, E. Plotkin, Automorphisms of the category of free Lie algebras, J. Algebra 282 (2004) 490–512]. As a corollary, semi-inner character of all automorphisms of the category of free Lie algebras over R 1 MF -domains is obtained. Relations between categorical and geometrical equivalence of Lie algebras over R 1 MF -domains are clarified. The group Aut End F for the variety of m-nilpotent associative algebras over R 1 MF -domains is described. As a consequence, a complete description of the group of automorphisms of the full matrix semigroup of n × n matrices over R 1 MF -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 End F are quasi-inner. The results obtained generalize the previous studies of various special cases of varieties of linear algebras over infinite fields.