Canadian Mathematical Society
www.cms.math.ca
|          |    |           |    |
|----------|----|-----------|----|
|          | |  |           | |  |
| Site map | |  | CMS store |    |

location:            Publications → journals → CMB
Abstract view

# Some Questions about Semisimple Lie Groups Originating in Matrix Theory

Read article
[PDF: 177KB]
http://dx.doi.org/10.4153/CMB-2003-035-1
Canad. Math. Bull. 46(2003), 332-343
Published:2003-09-01
Printed: Sep 2003
• Dragomir Z. Đoković
• Tin-Yau Tam
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:

Format:  HTML  LaTeX  MathJax  PDF  PostScript

## Abstract

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.

MSC Classifications:  15A45 - Miscellaneous inequalities involving matrices 20G20 - Linear algebraic groups over the reals, the complexes, the quaternions 22E60 - Lie algebras of Lie groups {For the algebraic theory of Lie algebras, see 17Bxx}
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 13, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.7313686609268188, "perplexity_flag": "middle"}