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Some Questions about Semisimple Lie Groups Originating in Matrix Theory

Published online by Cambridge University Press:  20 November 2018

Dragomir Ž. Ðoković  and
Tin-Yau Tam
Dragomir Ž. Ðoković
Affiliation:
Department of Pure Mathematics University of Waterloo Waterloo, Ontario N2L 3G1, email: djokovic@uwaterloo.ca
Tin-Yau Tam
Affiliation:
Department of Mathematics Auburn University AL 36849—5310 USA, email: tamtiny@auburn.edu
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Abstract

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We generalize the well-known result that a square traceless complex matrix is unitarily similar to a matrix with zero diagonal to arbitrary connected semisimple complex Lie groups $G$ and their Lie algebras $\mathfrak{g}$ under the action of a maximal compact subgroup $K$ of $G$. We also introduce a natural partial order on $\mathfrak{g}:\,x\,\le y$ if $f(K\,\cdot \,x)\,\subseteq \,f(K\,\cdot \,y)$ for all $f\,\in \,{{\mathfrak{g}}^{*}}$, the complex dual of $\mathfrak{g}$. This partial order is $K$-invariant and induces a partial order on the orbit space $\mathfrak{g}/K$. We prove that, under some restrictions on $\mathfrak{g}$, the set $f(K\,\cdot \,x)$ is star-shaped with respect to the origin.

Keywords

15A4520G2022E60
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 46 , Issue 3 , 01 September 2003 , pp. 332 - 343
Copyright
Copyright © Canadian Mathematical Society 2003

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