Automorphisms of real semisimple Lie algebras and their restricted root systems

TL;DR

This paper proves that automorphisms of the restricted root system of a real semisimple Lie algebra can be lifted to automorphisms of the algebra itself, with applications to symmetric spaces.

Contribution

It establishes a correspondence between automorphisms of restricted root systems and automorphisms of the Lie algebra, extending understanding of symmetry structures.

Findings

Automorphisms of restricted root systems can be lifted to Lie algebra automorphisms.

Results apply to automorphisms of Dynkin diagrams of restricted root systems.

Implications for the theory of symmetric spaces of noncompact type.

Abstract

We prove that every automorphism of the restricted root system of a real semisimple Lie algebra -- when defined properly -- can be lifted to an automorphism of that Lie algebra. In particular, this can be applied to automorphisms of the Dynkin diagram of the restricted root system. We also discuss some applications of this result to the theory of symmetric spaces of noncompact type.