Some Questions about Semisimple Lie Groups Originating in Matrix Theory

[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.

[学科分类] 数学（综合）

[关键词] Jacobian Conjecture;injectivity;Monge--Ampère equation