Some Questions about Semisimple Lie Groups Originating in Matrix Theory

Canadian mathematical bulletin

关键词: Jacobian Conjecture;    injectivity;    Monge--Ampère equation;   

DOI  :  10.4153/CMB-2003-035-1

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

We generalize the well-known result that a square traceless complexmatrix is unitarily similar to a matrix with zero diagonal toarbitrary connected semisimple complex Lie groups $G$ and their Liealgebras $mathfrak{g}$ under the action of a maximal compact subgroup$K$ of $G$. We also introduce a natural partial order on$mathfrak{g}$: $xle y$ if $f(Kcdot x) subseteq f(Kcdot y)$ forall $fin mathfrak{g}^*$, the complex dual of $mathfrak{g}$. Thispartial order is $K$-invariant and induces a partial order on theorbit space $mathfrak{g}/K$. We prove that, under some restrictionson $mathfrak{g}$, the set $f(Kcdot x)$ is star-shaped with respectto the origin.

【 授权许可】

Unknown