Endogenies and Linearisation

TL;DR

This paper demonstrates that certain algebraic actions of commuting endomorphism rings on bi-modules can be simplified into linear actions over a definable field, under specific conditions.

Contribution

It introduces conditions under which the action of commuting endogenies on bi-modules linearizes into vector spaces over a definable field.

Findings

Actions of two infinite commuting invariant rings linearize into vector spaces.

Linearization holds for strongly commuting endogenies with finite kernel.

Provides a framework for understanding algebraic actions via linearization.

Abstract

We show that the action of two infinite commuting invariant rings of endomorphisms of a finite-dimensional virtually connected irreducible bi-module linearizes into a vector space over a definable field. The same holds if the action is merely by strongly commuting endogenies, modulo some finite katakernel.