Lie rings in finite-dimensional theories

TL;DR

This paper classifies finite-dimensional Lie rings within certain logical frameworks, verifies a version of the Cherlin-Zilber Conjecture in characteristic zero, and characterizes actions and definable envelopes for specific Lie ring classes.

Contribution

It extends classification results for Lie rings to finite-dimensional theories, including NIP and connected cases, and verifies key conjectures and properties in this setting.

Findings

Classification of Lie rings up to dimension four in NIP or connected case

Verification of a version of the Cherlin-Zilber Conjecture in characteristic zero

Existence of definable envelopes for nilpotent and soluble Lie rings

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.