Geodesic
Some Questions about Semisimple Lie Groups Originating in Matrix Theory
Canadian mathematical bulletin, Tome 46 (2003) no. 3, pp. 332-343

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-2003-035-1
Mots-clés : 15A45, 20G20, 22E60 
Ðoković, Dragomir Ž.; Tam, Tin-Yau. Some Questions about Semisimple Lie Groups Originating in Matrix Theory. Canadian mathematical bulletin, Tome 46 (2003) no. 3, pp. 332-343. doi: 10.4153/CMB-2003-035-1
@article{10_4153_CMB_2003_035_1,
     author = {{\DH}okovi\'c, Dragomir \v{Z}. and Tam, Tin-Yau},
     title = {Some {Questions} about {Semisimple} {Lie} {Groups} {Originating} in {Matrix} {Theory}},
     journal = {Canadian mathematical bulletin},
     pages = {332--343},
     year = {2003},
     volume = {46},
     number = {3},
     doi = {10.4153/CMB-2003-035-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2003-035-1/}
}
TY  - JOUR
AU  - Ðoković, Dragomir Ž.
AU  - Tam, Tin-Yau
TI  - Some Questions about Semisimple Lie Groups Originating in Matrix Theory
JO  - Canadian mathematical bulletin
PY  - 2003
SP  - 332
EP  - 343
VL  - 46
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2003-035-1/
DO  - 10.4153/CMB-2003-035-1
ID  - 10_4153_CMB_2003_035_1
ER  - 
%0 Journal Article
%A Ðoković, Dragomir Ž.
%A Tam, Tin-Yau
%T Some Questions about Semisimple Lie Groups Originating in Matrix Theory
%J Canadian mathematical bulletin
%D 2003
%P 332-343
%V 46
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2003-035-1/
%R 10.4153/CMB-2003-035-1
%F 10_4153_CMB_2003_035_1

[1] [1] Au-Yeung, Y. H. and Sing, F. Y., A remark on the generalized numerical range of a normal matrix. Glasgow Math. J. 18 (1977), 179–180. Google Scholar

[2] [2] Bourbaki, N., Groupes et algèbres de Lie, Chapitres 4, 5 et 6. Hermann, Paris, 1968. Google Scholar

[3] [3] Bourbaki, N., Groupes et algèbres de Lie, Chapitres 7 et 8. Hermann, Paris, 1975. Google Scholar

[4] [4] Cheung, W. S. and Tsing, N. K., The C-numerical range of matrices is star-shaped. Linear and Multilinear Algebra 41 (1996), 245–250. Google Scholar

[5] [5] Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York, 1978. Google Scholar

[6] [6] Horn, R. A. and Johnson, C. R., Topics in Matrix Analysis. Cambridge University Press, 1991. Google Scholar

[7] [7] Humphreys, J. E., Introduction to Lie Algebras and Representation Theory. Springer-Verlag, New York 1972. Google Scholar

[8] [8] Li, C. K., Some convexity theorems for the generalized numerical ranges. Linear and Multilinear Algebra 40 (1996), 235–240. Google Scholar

[9] [9] Murnaghan, F. D., On the unitary invariants of a square matrix. Anais Acad. Brasil. Ci. 26 (1954), 1–7. Google Scholar

[10] [10] Nakazato, H., The C-numerical range of a 2 × 2 matrix. Sci. Rep. Hirosaki Univ. 41 (1994), 197–206. Google Scholar

[11] [11] Sibirskiĭ, K. S., The Algebraic Invariants of Differential Equations and Matrices. (Russian) Stiintsa, Kishinev, 1976. Google Scholar

[12] [12] Tam, T. Y., On the shape of numerical range associated with Lie groups. Taiwanese J. Math. 5 (2001), 497–506. Google Scholar

Cité par Sources :