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$5 \times 5$-graded Lie algebras, cubic norm structures and quadrangular algebras

Tom De Medts, Jeroen Meulewaeter

TL;DR

This work investigates simple Lie algebras generated by extremal elements over arbitrary fields and develops two parametrization paradigms for their structure. It shows that if the extremal geometry contains lines, the algebra admits a $5 imes5$-grading indexed by a cubic norm structure (via a twin cubic norm structure on $J$ and $J'$); under suitable field-extension and symplectic-pair hypotheses, a $5 imes5$-grading is parametrizable by a quadrangular algebra. A central technical advance is the introduction of $l$-exponential automorphisms that remain meaningful in characteristic $2$ and $3$, enabling a unified treatment of root-grading data without characteristic restrictions. These results link extremal Lie algebras to the algebraic frameworks of cubic norm structures and quadrangular algebras, shedding light on the underlying geometry of Moufang polygons and hexagons and providing tools for constructing and reconstructing the Lie algebras from these exceptional algebraic gadgets.

Abstract

We study simple Lie algebras generated by extremal elements, over arbitrary fields of arbitrary characteristic. We show: (1) If the extremal geometry contains lines, then the Lie algebra admits a $5 \times 5$-grading that can be parametrized by a cubic norm structure; (2) If there exists a field extension of degree at most $2$ such that the extremal geometry over that field extension contains lines, and in addition, there exist symplectic pairs of extremal elements, then the Lie algebra admits a $5 \times 5$-grading that can be parametrized by a quadrangular algebra. One of our key tools is a new definition of exponential maps that makes sense even over fields of characteristic $2$ and $3$, which ought to be interesting in its own right.

$5 \times 5$-graded Lie algebras, cubic norm structures and quadrangular algebras

TL;DR

This work investigates simple Lie algebras generated by extremal elements over arbitrary fields and develops two parametrization paradigms for their structure. It shows that if the extremal geometry contains lines, the algebra admits a -grading indexed by a cubic norm structure (via a twin cubic norm structure on and ); under suitable field-extension and symplectic-pair hypotheses, a -grading is parametrizable by a quadrangular algebra. A central technical advance is the introduction of -exponential automorphisms that remain meaningful in characteristic and , enabling a unified treatment of root-grading data without characteristic restrictions. These results link extremal Lie algebras to the algebraic frameworks of cubic norm structures and quadrangular algebras, shedding light on the underlying geometry of Moufang polygons and hexagons and providing tools for constructing and reconstructing the Lie algebras from these exceptional algebraic gadgets.

Abstract

We study simple Lie algebras generated by extremal elements, over arbitrary fields of arbitrary characteristic. We show: (1) If the extremal geometry contains lines, then the Lie algebra admits a -grading that can be parametrized by a cubic norm structure; (2) If there exists a field extension of degree at most such that the extremal geometry over that field extension contains lines, and in addition, there exist symplectic pairs of extremal elements, then the Lie algebra admits a -grading that can be parametrized by a quadrangular algebra. One of our key tools is a new definition of exponential maps that makes sense even over fields of characteristic and , which ought to be interesting in its own right.
Paper Structure (11 sections, 91 theorems, 253 equations, 2 figures, 4 tables)

This paper contains 11 sections, 91 theorems, 253 equations, 2 figures, 4 tables.

Table of Contents

  1. Preliminaries
  2. Lie algebras
  3. Cubic norm structures
  4. Quadrangular algebras
  5. Gradings from extremal elements
  6. $l$-exponential automorphisms
  7. Extremal geometry with lines --- recovering a cubic norm structure
  8. Extremal geometry with symplectic pairs --- recovering a quadrangular algebra
  9. Tables
  10. Cubic norm structures
  11. Quadrangular algebras

Key Result

Theorem A

Let $L$ be a simple Lie algebra. Assume that for some Galois extension $k'/k$ with $k'\neq\mathbb F_2$, $L_{k'} := L \otimes_k k'$ is a simple Lie algebra generated by its pure extremal elements such that the extremal geometry contains lines. Consider a hyperbolic pair $(x,y)$ of extremal elements a where $q_{\alpha},n_{\alpha},v_{\alpha} \colon L \to L$ are maps with for all $i\in [-2,2]$.

Figures (2)

  • Figure 1: Intersecting gradings for cubic norm structures
  • Figure 2: Intersecting gradings for quadrangular algebras

Theorems & Definitions (221)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 1.1
  • Definition 1.2: Cohen2006
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5: Cohen2006
  • Lemma 1.6: Cohen2006
  • Proposition 1.7: Cohen2006
  • ...and 211 more