On the Component Group of a Real Algebraic Group

Dmitry A. Timashev

Informatics and Automation, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 2, Tome 318 (2022), pp. 193-203

Résumé

For a connected linear algebraic group $G$ defined over $\mathbb R$, we compute the component group $\pi _0G(\mathbb R)$ of the real Lie group $G(\mathbb R)$ in terms of a maximal split torus $T_{\mathrm{s}}\subseteq G$. In particular, we recover a theorem of Matsumoto (1964) that each connected component of $G(\mathbb R)$ intersects $T_{\mathrm{s}}(\mathbb R)$. We provide explicit elements of $T_{\mathrm{s}}(\mathbb R)$ 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.

Détail

Keywords: real algebraic group, component group, split torus, real Galois cohomology.