Automorphisms and derivations of affine commutative and PI-algebras

TL;DR

This paper extends classical theorems to the automorphisms and derivations of finitely generated associative commutative algebras, establishing new analogs of known results in algebraic group theory and Lie algebra theory.

Contribution

It introduces novel analogs of Selberg's, Engel's, and Burnside-Schur theorems for automorphism groups and derivation subalgebras of finitely generated associative commutative algebras.

Findings

Analog of Selberg's theorem for automorphism groups

Analog of Engel's theorem for derivation subalgebras

Analog of Burnside-Schur theorem on torsion subgroups

Abstract

We prove analogs of A.~Selberg's result for finitely generated subgroups of $\text{Aut}(A)$ and of Engel's theorem for subalgebras of $\text{Der}(A)$ for a finitely generated associative commutative algebra $A$ over an associative commutative ring. We prove also an analog of the theorem of W.~Burnside and I.~Schur about locally finiteness of torsion subgroups of $\text{Aut}(A)$.