论文标题
在一个实际还原组的组件组上
On the component group of a real reductive group
论文作者
论文摘要
对于在$ \ mathbb {r} $上定义的连接的线性代数$ g $，我们计算真实lie组$ g（\ mathbb {r}）$的组件组$π_0g（\ mathbb {r}）$，按maximal split $ t _ q _ q $ q cy（\ mathbb {r}）$。特别是，我们恢复了Matsumoto（1964）的定理，即$ g（\ Mathbb {r}）的每个连接组件$ intersects $ t _ {\ text {s}}}（\ mathbb {r}）$。我们提供$ t _ {\ text {s}}}（\ Mathbb {r}）$的明确元素，该元素表示$ g（\ Mathbb {r}）$的所有连接组件。该计算基于代数组的实际基因座的结构结果和GALOIS协同学方法。
For a connected linear algebraic group $G$ defined over $\mathbb{R}$, we compute the component group $π_0G(\mathbb{R})$ of the real Lie group $G(\mathbb{R})$ in terms of a maximal split torus $T_{\text{s}}\subseteq G$. In particular, we recover a theorem of Matsumoto (1964) that each connected component of $G(\mathbb{R})$ intersects $T_{\text{s}}(\mathbb{R})$. We provide explicit elements of $T_{\text{s}}(\mathbb{R})$ which represent all connected components of $G(\mathbb{R})$. The computation is based on structure results for real loci of algebraic groups and on methods of Galois cohomology.
