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Lie rings in finite-dimensional theories

Moreno Invitti

TL;DR

This work extends Cherlin–Zilber-type insights from finite Morley rank to finite-dimensional theories for definable Lie rings, deriving a comprehensive structural picture up to dimension four and char $>3$, including a dimensional- and NIP-aware classification of connected Lie rings. It develops a robust almost Lie ring framework (almost centralizers, almost ideals, absolute simplicity) and linearization tools that yield $K$-linear actions over definable fields, even in non-virtually-connected settings, and uses these to prove a dimensional, characteristic-zero version of the log-CZ conjecture. The paper further establishes definable envelopes for soluble and nilpotent subrings, resulting in definability of the Fitting and Radical ideals and a detailed analysis of their almost variants within dimensional theories and the $ ilde{ rak M}_c$-context. Through these methods, it provides a coherent pathway to classify stable and NIP Lie rings of finite dimension, and to address questions about envelopes, linearization, and definable simplicity in a broad logical landscape. Its findings illuminate how dimension, model-theoretic tameness, and Lie-theoretic structure interact to yield definability, linearization, and envelope phenomena across both characteristic regimes.

Abstract

We study Lie rings definable in a finite-dimensional theory, extending the results for the finite Morley rank case. In particular, we prove a classification of Lie rings of dimension up to four in the NIP or connected case. In characteristic $0$, we verify a version of the Cherlin-Zilber Conjecture. Moreover, we characterize the actions of some classes, namely abelian, nilpotent and soluble, of Lie rings of finite dimension. Finally, we show the existence of definable envelopes for nilpotent and soluble Lie rings. These results are used to verify that the Fitting and the Radical ideal of a Lie ring of finite dimension are both definable and respectively nilpotent and soluble.

Lie rings in finite-dimensional theories

TL;DR

This work extends Cherlin–Zilber-type insights from finite Morley rank to finite-dimensional theories for definable Lie rings, deriving a comprehensive structural picture up to dimension four and char , including a dimensional- and NIP-aware classification of connected Lie rings. It develops a robust almost Lie ring framework (almost centralizers, almost ideals, absolute simplicity) and linearization tools that yield -linear actions over definable fields, even in non-virtually-connected settings, and uses these to prove a dimensional, characteristic-zero version of the log-CZ conjecture. The paper further establishes definable envelopes for soluble and nilpotent subrings, resulting in definability of the Fitting and Radical ideals and a detailed analysis of their almost variants within dimensional theories and the -context. Through these methods, it provides a coherent pathway to classify stable and NIP Lie rings of finite dimension, and to address questions about envelopes, linearization, and definable simplicity in a broad logical landscape. Its findings illuminate how dimension, model-theoretic tameness, and Lie-theoretic structure interact to yield definability, linearization, and envelope phenomena across both characteristic regimes.

Abstract

We study Lie rings definable in a finite-dimensional theory, extending the results for the finite Morley rank case. In particular, we prove a classification of Lie rings of dimension up to four in the NIP or connected case. In characteristic , we verify a version of the Cherlin-Zilber Conjecture. Moreover, we characterize the actions of some classes, namely abelian, nilpotent and soluble, of Lie rings of finite dimension. Finally, we show the existence of definable envelopes for nilpotent and soluble Lie rings. These results are used to verify that the Fitting and the Radical ideal of a Lie ring of finite dimension are both definable and respectively nilpotent and soluble.

Paper Structure

This paper contains 30 sections, 102 theorems, 60 equations.

Table of Contents

  1. Preliminaries
  2. Finite-dimensional groups
  3. Lie rings
  4. Connected simple Lie rings
  5. Characterization of Lie rings of small dimension
  6. Dimension 1
  7. Dimension 2
  8. Dimension 3
  9. Dimension 4
  10. Almost Lie ring theory
  11. Almost centraliser
  12. Almost ideals and absolute simplicity
  13. Linearising in the non virtually connected case
  14. Dimensional Lie rings of characteristic 0
  15. NIP Lie rings
  16. ...and 15 more sections

Key Result

Theorem 1

Let $\mathfrak{g}$ be a connected definable Lie ring of finite Morley rank. Then,

Theorems & Definitions (212)

  • Conjecture
  • Theorem 1
  • Theorem 2
  • Definition
  • Conjecture
  • Theorem
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • ...and 202 more