Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semisimple algebraic groups over real closed fields

Raphael Appenzeller

Abstract

We give a self-contained introduction to linear algebraic and semialgebraic groups over real closed fields, and we generalize several key results about semisimple Lie groups to algebraic and semialgebraic groups over real closed fields. We prove that a torus in a semisimple algebraic group is maximal $\mathbb{R}$-split if and only if it is maximal $\mathbb{F}$-split for real closed fields $\mathbb{F}$. For the $\mathbb{F}$-points we formulate and prove the Iwasawa-decomposition $KAU$, the Cartan-decomposition $KAK$ and the Bruhat-decomposition $BWB$. For unipotent subgroups we prove the Baker-Campbell-Hausdorff formula, facilitating the analysis of root groups. We give a proof of the Jacobson-Morozov Lemma about subgroups whose Lie algebra is isomorphic to $\mathfrak{sl}_2$ for algebraic groups and a version for the $\mathbb{F}$-points, when the root system is reduced. We describe the rank 1 subgroups which are the semisimple parts of Levi-subgroups. We prove a semialgebraic version of Kostant's convexity theorem. The main tool used is a model theoretic transfer principle that follows from the Tarski-Seidenberg theorem.

Semisimple algebraic groups over real closed fields

Abstract

We give a self-contained introduction to linear algebraic and semialgebraic groups over real closed fields, and we generalize several key results about semisimple Lie groups to algebraic and semialgebraic groups over real closed fields. We prove that a torus in a semisimple algebraic group is maximal -split if and only if it is maximal -split for real closed fields . For the -points we formulate and prove the Iwasawa-decomposition , the Cartan-decomposition and the Bruhat-decomposition . For unipotent subgroups we prove the Baker-Campbell-Hausdorff formula, facilitating the analysis of root groups. We give a proof of the Jacobson-Morozov Lemma about subgroups whose Lie algebra is isomorphic to for algebraic groups and a version for the -points, when the root system is reduced. We describe the rank 1 subgroups which are the semisimple parts of Levi-subgroups. We prove a semialgebraic version of Kostant's convexity theorem. The main tool used is a model theoretic transfer principle that follows from the Tarski-Seidenberg theorem.
Paper Structure (35 sections, 56 theorems, 206 equations, 4 figures)

This paper contains 35 sections, 56 theorems, 206 equations, 4 figures.

Table of Contents

  1. Introduction and main results
  2. Applications
  3. Semialgebraic groups and o-minimality.
  4. Outline
  5. Acknowledgements
  6. Real closed fields
  7. The transfer principle
  8. Examples of real closed fields
  9. Linear algebraic groups
  10. Definitions
  11. Examples
  12. Morphisms and tori
  13. Examples
  14. The Lie Algebra and the adjoint representation
  15. The Lie Algebra and the adjoint representation
  16. ...and 20 more sections

Key Result

Theorem 2.1

(Transfer principle, BCR) Let $\mathbb{F}$ and $\mathbb{F} '$ be real closed fields. Let $\varphi$ be a sentence with parameters in $\mathbb{F} \cap \mathbb{F} '$. Then $\varphi$ is true for $\mathbb{F}$ if and only if $\varphi$ is true for $\mathbb{F} '$, formally $\mathbb{F} \models \varphi \iff

Figures (4)

  • Figure 1: Root system of type $A_2$ associated to $\operatorname{SL}_3$. The convex cone $\mathfrak{a}_p$ (orange stripes) can be viewed as spanned by the $H_{\alpha_i}$ or as the intersection of the half-spaces defined by the primitive vectors $e_i$.
  • Figure 2: Root system of type $A_2$ associated to $\operatorname{SL}_3$. The convex set in Kostant's convexity Theorem \ref{['thm:kostant_F']} defined by inequalities is illustrated in purple.
  • Figure 3: The root system of type $\operatorname{G}_2$ with basis $\delta_1,\delta_2$, the coroots $\delta_1^\vee, \delta_2^\vee$ corresponding to the coroots $x_1,x_2$ and the elements $\gamma_1, \gamma_2$ spanning the Weyl chamber $\overline{(\mathfrak{a}^\star)}^+$. The element $\eta^+ := \sum_{\alpha >0} \alpha$ from Lemma \ref{['lem:kostant_gammaalpha']} lies in $\overline{(\mathfrak{a}^\star)}^+$, while the other candidate $\eta := \delta_1 + \delta_2$ does not.
  • Figure :

Theorems & Definitions (92)

  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.2: 8.5 in Bor
  • Theorem 3.3: 8.4 in Bor
  • Theorem 3.4: 11.3 and 20.9 in Bor
  • Proposition 3.5
  • Theorem 3.6: 21.6 Bor
  • Theorem 3.7
  • Lemma 4.1
  • ...and 82 more