On realizations of the Lie groups $ G_{2,\boldmath\scriptstyle{H}},F_{4,\boldmath\scriptstyle{H}},E_{6,\boldmath\scriptstyle{H}},E_{7,\boldmath\scriptstyle{H}},E_{8,\boldmath\scriptstyle{H}} $, second edition

TL;DR

This paper explores alternative constructions of exceptional compact Lie groups by replacing the Cayley algebra with the quaternion algebra, aiming to understand their structure through these new realizations.

Contribution

It introduces new realizations of the exceptional Lie groups using quaternion algebra instead of Cayley algebra, providing insights into their structure.

Findings

New group realizations using quaternion algebra

Structural descriptions of the groups

Comparison with traditional Cayley algebra-based groups

Abstract

In order to define the exceptional compact Lie groups $G_2,F_4,E_6,E_7,E_8$, we usually use the Cayley algebra $\mathfrak{C}$ or its complexification $\mathfrak{C}^C$. In the present article, we consider replacing the Cayley algebra $\mathfrak{C}$ with the field of quaternion numbers $\boldmath{H}$ in the definition of the groups above, and these groups are denoted as in title above. Our aim is to determine the structure of these groups. We call realization to determine the structure of the groups.