Soluble Lie rings of finite Morley rank

TL;DR

This paper extends classical Lie theory to the setting of soluble Lie rings of finite Morley rank, proving a Morley rank version of the Lie-Kolchin-Malcev theorem and classifying certain Lie ring actions.

Contribution

It introduces a Morley rank analogue of the Lie-Kolchin-Malcev theorem and classifies Lie ring actions on modules of Morley rank 2 in characteristic not 2 or 3.

Findings

Finite Morley rank version of Lie-Kolchin-Malcev theorem established

Classification of Lie ring actions on Morley rank 2 modules achieved

Linearisation results for soluble Lie rings of finite Morley rank obtained

Abstract

We do two things. 1. As a corollary to a stronger linearisation result (Theorem A), we prove the finite Morley rank version of the Lie-Kolchin-Malcev theorem on Lie algebras (Corollary A2). 2. We classify Lie ring actions on modules of characteristic not 2, 3 and Morley rank 2 (Theorem B).