论文标题
$ \ mathfrak {sl}（2，\ mathbb {r}）$的伴随轨道及其几何形状
Adjoint orbits of $\mathfrak{sl}(2,\mathbb{R})$ and their geometry
论文作者
论文摘要
令$ \ mathrm {sl}（n，\ mathbb {r}）$为特殊线性组和$ \ mathfrak {sl}（n，\ mathbb {r}）$它的lie代数。我们研究了最简单的非平凡情况下与伴随轨道相关的几何属性，即$ \ mathfrak {sl}（2，\ mathbb {r}）$的几何特性。特别是，我们表明只有三种可能性：伴随轨道是单片的双曲体，或者是两片曲折的双曲体，或者是锥体。 此外，我们引入了特定的潜力，并研究了相应的梯度矢量场及其动力学时，当我们限制在伴随轨道上时。我们通过描述来自众所周知的Kirillov-Kostant-Souriau souriau simbletectic形式的隔离轨道上的互合结构来结束。
Let $\mathrm{SL}(n,\mathbb{R})$ be the special linear group and $\mathfrak{sl}(n,\mathbb{R})$ its Lie algebra. We study geometric properties associated to the adjoint orbits in the simplest non-trivial case, namely, those of $\mathfrak{sl}(2,\mathbb{R})$. In particular, we show that just three possibilities arise: either the adjoint orbit is a one-sheeted hyperboloid, or a two-sheeted hyperboloid, or else a cone. In addition, we introduce a specific potential and study the corresponding gradient vector field and its dynamics when we restricted to the adjoint orbit. We conclude by describing the symplectic structure on these adjoint orbits coming from the well known Kirillov-Kostant-Souriau symplectic form on coadjoint orbits.
