The Iwasawa decomposition and the Bruhat decomposition of the automorphism group on certain exceptional Jordan algebra

TL;DR

This paper explicitly derives various decompositions, including Iwasawa and Bruhat, for the automorphism group of a specific exceptional Jordan algebra, enhancing understanding of its structure.

Contribution

It provides explicit formulas for multiple decompositions of the automorphism group of a real form of a complex simple Jordan algebra, which was previously not fully detailed.

Findings

Explicit Iwasawa decomposition for F4(-20)

Derivation of Bruhat and Gauss decompositions

Application of orbit types to decompose automorphism group

Abstract

Let $\mathcal{J}^1$ be the real form of a complex simple Jordan algebra such that the automorphism group is $\mathrm{F}_{4(-20)}$. By using some orbit types of $\mathrm{F}_{4(-20)}$ on $\mathcal{J}^1$, for $\mathrm{F}_{4(-20)}$, explicitly, we give the Iwasawa decomposition, the Oshima--Sekiguchi's $K_{\epsilon}-$Iwasawa decomposition, the Matsuki decomposition, and the Bruhat and Gauss decompositions.