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Homogeneous Freudenthal algebras and the first Tits construction

Holger P. Petersson, Maneesh Thakur

Abstract

Freudenthal algebras over a field are basically the same as Jordan algebras of degree $3$ remaining simple under all base field extensions. These algebras are intimately linked, via their automorphism groups and structure groups, to simple algebraic groups over arbitrary fields. Our main concern here will be the question of when these algebras are homogeneous in the sense that all their Jordan isotopes are isomorphic. We answer this question by presenting various necessary and sufficient conditions for homogeneity and by connecting it with the first Tits construction of cubic Jordan algebras, most notably through investigating Freudenthal division algebras over complete fields under a discrete valuation. We also study the first Tits construction in its own right by producing a local version of it, deriving a local-global principle, and by connecting it with the embeddibility of certain rank-$2$-tori into the automorphism group scheme of an Albert division algebra.

Homogeneous Freudenthal algebras and the first Tits construction

Abstract

Freudenthal algebras over a field are basically the same as Jordan algebras of degree remaining simple under all base field extensions. These algebras are intimately linked, via their automorphism groups and structure groups, to simple algebraic groups over arbitrary fields. Our main concern here will be the question of when these algebras are homogeneous in the sense that all their Jordan isotopes are isomorphic. We answer this question by presenting various necessary and sufficient conditions for homogeneity and by connecting it with the first Tits construction of cubic Jordan algebras, most notably through investigating Freudenthal division algebras over complete fields under a discrete valuation. We also study the first Tits construction in its own right by producing a local version of it, deriving a local-global principle, and by connecting it with the embeddibility of certain rank--tori into the automorphism group scheme of an Albert division algebra.
Paper Structure (28 sections, 49 theorems, 49 equations)

This paper contains 28 sections, 49 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Proper Freudenthal algebras and the Tits constructions
  3. Homogeneous Jordan algebras
  4. Cubic Jordan algebras
  5. Proper reduced Freudenthal algebras
  6. The first Tits construction
  7. The second Tits construction
  8. Invariants
  9. Reduced models and the coordinate dimension
  10. The invariants $\bmod\,2$
  11. Invariants: Freudenthal algebras of dimension $9$
  12. Isotopes and distinguished involutions
  13. Invariants: Albert algebras
  14. Subalgebras arising from the first Tits construction
  15. The first Tits construction at a separable cubic subfield
  16. ...and 13 more sections

Key Result

Proposition 2.3

Let $C$ be a composition $k$-algebra, $J := \mathop{\mathrm{Her}}\nolimits_3(C)$ and With $p := \sum \gamma_ie_{ii} \in J^\times$, the assignment for $\xi_i \in k$, $u_i \in C$, $1\le i\le 3$, defines an isomorphism $\varphi$ from $\mathop{\mathrm{Her}}\nolimits_3(C,\Gamma)$ onto the isotope $J^{(p^{-1})}$.

Theorems & Definitions (94)

  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • Corollary 2.5
  • proof
  • Proposition 2.8
  • proof
  • Proposition 2.9
  • Theorem 2.10
  • ...and 84 more