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From cubic norm pairs to $G_2$- and $F_4$-graded groups and Lie algebras

Tom De Medts, Torben Wiedemann

Abstract

We construct Lie algebras arising from cubic norm pairs over arbitrary commutative base rings. Such Lie algebras admit a grading by a root system of type $G_2$, and when the cubic norm pair is a cubic Jordan matrix algebra, the $G_2$-grading can be further refined to an $F_4$-grading. We then use these Lie algebras and their gradings to construct corresponding root graded groups. Along the way, we produce many results providing detailed information about the structure of these Lie algebras and groups.

From cubic norm pairs to $G_2$- and $F_4$-graded groups and Lie algebras

Abstract

We construct Lie algebras arising from cubic norm pairs over arbitrary commutative base rings. Such Lie algebras admit a grading by a root system of type , and when the cubic norm pair is a cubic Jordan matrix algebra, the -grading can be further refined to an -grading. We then use these Lie algebras and their gradings to construct corresponding root graded groups. Along the way, we produce many results providing detailed information about the structure of these Lie algebras and groups.
Paper Structure (30 sections, 108 theorems, 358 equations, 3 figures, 2 tables)

This paper contains 30 sections, 108 theorems, 358 equations, 3 figures, 2 tables.

Table of Contents

  1. Root graded Lie algebras and groups
  2. Root gradings
  3. F4-graded groups
  4. Cubic norm pairs
  5. Definition and identities
  6. Linear cubic norm pairs
  7. The Tits--Kantor--Koecher Lie algebra for cubic norm pairs
  8. Constructing the Lie algebra
  9. Constructing the bracket
  10. Verification of the Jacobi identity
  11. Functoriality and reflections
  12. The ideal structure of L
  13. Exponential maps for the G2-grading
  14. General properties of exponential maps
  15. Definition of exponential maps in L
  16. ...and 15 more sections

Key Result

Theorem A

Let $(J,J')$ be a cubic norm pair over a commutative ring $k$. Then there exists a $G_2$-graded Lie algebra $L=L(J,J')$ whose root spaces $L_\alpha$ for $\alpha \in G_2$ are isomorphic to $k$, $J$ or $J'$. Further, the construction of $L$ is functorial in $(J,J')$ (with respect to surjective homotop

Figures (3)

  • Figure 1: Construction of a $5 \times 5$-graded Lie algebra
  • Figure 2: The Tits index $(F_4, \{\mathop{\mathrm{id}}\nolimits\}, \{3,4\})$, which we refer to as "$F_4 \to G_2$".
  • Figure 3: Intersecting the four gradings

Theorems & Definitions (343)

  • Theorem A: \ref{['thm:G2 Lie']}, \ref{['le:reflections auto']}, \ref{['thm:L simple']}
  • Theorem B: \ref{['thm:G2 graded group']}
  • Theorem C: \ref{['thm:lie F4 grading']}, \ref{['thm:F4 graded group']}
  • Definition 1.2
  • Remark 1.3
  • Definition 1.5: WiedemannPhD
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8: WiedemannPhD
  • Definition 1.9
  • ...and 333 more