Topologically Conjugate Classifications of the Translation Actions on Compact Connected Lie Groups ${\rm SU}(2) \times T^n$

TL;DR

This paper classifies translation actions on the compact Lie group SU(2) x T^n using rotation vectors, providing a comprehensive topological conjugacy framework including algebraic and smooth conjugacies.

Contribution

It introduces a complete classification of left translation actions on SU(2) x T^n via rotation vectors, extending understanding of conjugacy in noncommutative compact Lie groups.

Findings

Complete topological conjugacy classification achieved

Rotation vectors effectively distinguish conjugacy classes

Connections between algebraic, smooth, and topological conjugacies established

Abstract

In this article, we focus on the left (translation) actions on noncommutative compact connected Lie groups ${\rm SU}(2) \times T^n$. We define the rotation vectors of the left actions induced by the elements in the maximal tori of ${\rm SU}(2) \times T^n$, and utilize rotation vectors to give the complete topologically conjugate classifications of left actions. Algebraic conjugacy and smooth conjugacy are also considered.