The full group of isometries of some compact Lie groups endowed with a bi-invariant metric

TL;DR

This paper characterizes the complete group of symmetries for certain compact Lie groups with bi-invariant metrics, including simple, SO(4), and U(n), revealing their geometric symmetry structures.

Contribution

It provides a comprehensive description of the isometry groups for specific classes of compact Lie groups with bi-invariant metrics, extending understanding of their geometric symmetries.

Findings

Full isometry group of simple compact Lie groups identified

Isometry group of SO(4) explicitly described

Isometry group of U(n) explicitly described

Abstract

We describe the full group of isometries of absolutely simple, compact, connected real Lie groups, of SO(4) and of U(n), endowed with suitable bi-invariant Riemannian metrics.