Conjugacy and element-conjugacy of homomorphisms of compact Lie groups

Yingjue Fang, Gang Han and Binyong Sun

Vol. 283 (2016), No. 1, 75–83

DOI: 10.2140/pjm.2016.283.75

Abstract

Let $G$ be a connected compact Lie group. Among other things, we prove that the following are equivalent. (a) For all connected compact Lie groups $H$ and all continuous homomorphisms $\varphi ,{\varphi }^{\prime }:H\to G$, if $\varphi \left(h\right)$ and ${\varphi }^{\prime }\left(h\right)$ are conjugate in $G$ for all $h\in H$, then $\varphi$ and ${\varphi }^{\prime }$ are $G$-conjugate. (b) The Lie algebra of $G$ contains no simple ideal of type $D_n$ ($n\ge 4$), $E_6$, $E_7$, or $E_8$.

Keywords

element-conjugacy, conjugacy, compact Lie group

Mathematical Subject Classification 2010

Primary: 22E46

Milestones

Received: 29 September 2015

Revised: 13 January 2016

Accepted: 14 January 2016

Published: 14 June 2016