CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

Star-Shapedness and $K$-Orbits in Complex Semisimple Lie Algebras

  Published:2010-08-26
 Printed: Mar 2011
  • Wai-Shun Cheung,
    Department of Mathematics and Statistics, Auburn University, AL 36849--5310, U.S.A.
  • Tin-Yau Tam,
    Department of Mathematics and Statistics, Auburn University, AL 36849--5310, U.S.A.
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   HTML   LaTeX   MathJax   PDF  

Abstract

Given a complex semisimple Lie algebra $\mathfrak{g}=\mathfrak{k}+i\mathfrak{k}$ ($\mathfrak{k}$ is a compact real form of $\mathfrak{g}$), let $\pi\colon\mathfrak{g}\to \mathfrak{h}$ be the orthogonal projection (with respect to the Killing form) onto the Cartan subalgebra $\mathfrak{h}:=\mathfrak{t}+i\mathfrak{t}$, where $\mathfrak{t}$ is a maximal abelian subalgebra of $\mathfrak{k}$. Given $x\in \mathfrak{g}$, we consider $\pi(\mathop{\textrm{Ad}}(K) x)$, where $K$ is the analytic subgroup $G$ corresponding to $\mathfrak{k}$, and show that it is star-shaped. The result extends a result of Tsing. We also consider the generalized numerical range $f(\mathop{\textrm{Ad}}(K)x)$, where $f$ is a linear functional on $\mathfrak{g}$. We establish the star-shapedness of $f(\mathop{\textrm{Ad}}(K)x)$ for simple Lie algebras of type $B$.
MSC Classifications: 22E10, 17B20 show english descriptions General properties and structure of complex Lie groups [See also 32M05]
Simple, semisimple, reductive (super)algebras
22E10 - General properties and structure of complex Lie groups [See also 32M05]
17B20 - Simple, semisimple, reductive (super)algebras
 

© Canadian Mathematical Society, 2014 : http://www.cms.math.ca/