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Cartan subrings in soluble ranked Lie rings

Jules Tindzogho Ntsiri, Samuel Zamour

TL;DR

The paper addresses the existence and characterization of Cartan subrings in soluble ranked Lie rings of finite Morley rank, extending prior results for specific cases. It adapts Frécon's abnormal-subgroup framework to Lie rings by introducing def-abnormal subrings and links them to Cartan subrings via minimality and Engel theory, aided by Frattini and Fitting subring techniques. Under the key hypothesis $\operatorname{char}(\mathfrak{g}) > \operatorname{rk}(\mathfrak{g})$ and with $\mathfrak{g}'$ nilpotent, it proves both existence of Cartan subrings and a precise equivalence among minimal def-abnormal, Cartan, and Engel-minimal subrings, along with conjugacy results when $Z^{\circ}(\mathfrak{g})=0$. The work provides a cohesive Lie-ring analogue of Carter-type structure theory in finite Morley rank groups, clarifying how Cartan, Frattini, and Engel substructures interact in soluble ranked Lie rings.

Abstract

We prove the existence of Cartan subrings, i.e., self-normalizing nilpotent subrings in soluble ranked Lie rings.

Cartan subrings in soluble ranked Lie rings

TL;DR

The paper addresses the existence and characterization of Cartan subrings in soluble ranked Lie rings of finite Morley rank, extending prior results for specific cases. It adapts Frécon's abnormal-subgroup framework to Lie rings by introducing def-abnormal subrings and links them to Cartan subrings via minimality and Engel theory, aided by Frattini and Fitting subring techniques. Under the key hypothesis and with nilpotent, it proves both existence of Cartan subrings and a precise equivalence among minimal def-abnormal, Cartan, and Engel-minimal subrings, along with conjugacy results when . The work provides a cohesive Lie-ring analogue of Carter-type structure theory in finite Morley rank groups, clarifying how Cartan, Frattini, and Engel substructures interact in soluble ranked Lie rings.

Abstract

We prove the existence of Cartan subrings, i.e., self-normalizing nilpotent subrings in soluble ranked Lie rings.

Paper Structure

This paper contains 5 sections, 24 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. (Def)-Abnormal subrings
  4. Frattini subring
  5. Engel's subrings

Key Result

Lemma 2.4

Let $\mathfrak{g}$ be a connected soluble ranked Lie ring such that $\operatorname{char}(\mathfrak{g})> \operatorname{rk}(\mathfrak{g})$ and $V$ a $\mathfrak{g}$-minimal module on which $\mathfrak{g}$ acts non-trivially. Then $V$ is $\mathfrak{g}$-irreducible.

Theorems & Definitions (57)

  • Definition 1.1
  • Conjecture 1.2
  • Definition 2.1
  • Lemma 2.4
  • proof
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Proposition 3.1
  • proof
  • ...and 47 more