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Soluble Lie rings of finite Morley rank

Adrien Deloro, Jules Tindzogho Ntsiri

TL;DR

The paper establishes a finite Morley rank analogue of Lie theory for Lie rings by proving a linearisation theorem: if a Lie ring g of finite Morley rank has an irreducible, faithful module V with char V exceeding its dimension and contains an infinite abelian ideal a, then the action is definably linear over an infinite definable field K with a acting as scalars. It also provides a complete classification of 2‑dimensional definable, irreducible g‑modules, showing that over a definable field K the action is either scalar on V or arises from the standard representations of sl2(K) or gl2(K). The results yield structural corollaries on soluble Lie rings, including the existence of a largest connected nilpotent F∘(r) and nilpotence of the derived subring under a prime-exponent condition, and refine Lie-Kolchin-Malcev type linearisation to the Lie ring setting. Overall, the work extends model-theoretic algebra to finite Morley rank Lie rings, providing tools for linearising actions and classifying low‑dimensional representations, with implications for the broader study of Lie rings of finite Morley rank.

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).

Soluble Lie rings of finite Morley rank

TL;DR

The paper establishes a finite Morley rank analogue of Lie theory for Lie rings by proving a linearisation theorem: if a Lie ring g of finite Morley rank has an irreducible, faithful module V with char V exceeding its dimension and contains an infinite abelian ideal a, then the action is definably linear over an infinite definable field K with a acting as scalars. It also provides a complete classification of 2‑dimensional definable, irreducible g‑modules, showing that over a definable field K the action is either scalar on V or arises from the standard representations of sl2(K) or gl2(K). The results yield structural corollaries on soluble Lie rings, including the existence of a largest connected nilpotent F∘(r) and nilpotence of the derived subring under a prime-exponent condition, and refine Lie-Kolchin-Malcev type linearisation to the Lie ring setting. Overall, the work extends model-theoretic algebra to finite Morley rank Lie rings, providing tools for linearising actions and classifying low‑dimensional representations, with implications for the broader study of Lie rings of finite Morley rank.

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).
Paper Structure (10 sections, 4 theorems, 1 equation)

This paper contains 10 sections, 4 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Present results (for quick reference)
  3. Terminology and notation
  4. Linearisation, and Lie's theorem
  5. Linearisation theorem
  6. Corollaries
  7. Modules of Morley rank 2
  8. Proof of Theorem \ref{['t:module2']}
  9. Rewriting a classic
  10. Acknowledgements.

Key Result

Theorem A

Let $\mathfrak{g}$ be a Lie ring of finite Morley rank (not necessarily connected). Let $V$ be a definable, irreducible, faithful $\mathfrak{g}$-module. Suppose $\operatorname{char} V > \operatorname{dim} V$. Suppose that $\mathfrak{g}$ has an infinite abelian ideal $\mathfrak{a} \trianglelefteq \ma

Theorems & Definitions (20)

  • Theorem A
  • proof
  • proof
  • proof
  • proof
  • proof
  • Corollary A1
  • proof
  • Corollary A2
  • proof
  • ...and 10 more