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

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.

arXiv commentv3, 30 pages, 1 figure, major changes from v2 by separating section 9--14, reformatted, and the title changed to appear in Tsukuba J. of Math

References in corpus (2)

• Exceptional Lie groups
• The classification of orbits on certain exceptional Jordan algebra under the automorphism group