Exceptional Lie groups

TL;DR

This paper provides a concrete construction and classification of simply connected compact exceptional Lie groups, their involutive automorphisms, fixed point subgroups, and maximal subgroups, linking to symmetric spaces and non-compact forms.

Contribution

It offers an elementary, explicit construction of exceptional Lie groups and classifies their automorphisms and subgroups, connecting to symmetric space theory.

Findings

Constructed all simply connected compact exceptional Lie groups.

Classified involutive automorphisms and fixed point subgroups.

Determined structures of maximal subgroups of maximal rank.

Abstract

We describe simply connected compact exceptional simple Lie groups in very elementary way. We first construct all simply connected compact exceptional Lie groups G concretely. Next, we find all involutive automorphisms of G, and determine the group structures of the fixed points subgroup. They correspond to the classification of all irreducible compact symmetric spaces of exceptional type, and that they also correspond to classification of all non-compact exceptionalsimple Lie groups. Finally, we determined the group structures of the maximal subgroups of maximal rank. At any rate, we would like this book to be used in mathematics and physics.