We compute explicitly the group of connected components $\pi_0G(\mathbb{ R})$ of the real Lie group $G(\mathbb{ R})$ for an arbitrary (not necessarily linear) connected algebraic group $G$ defined over the field $\mathbb{ R}$ of real numbers. In particular, it turns out that $\pi_0G(\mathbb{ R})$ is always an elementary Abelian 2-group. The result looks most transparent in the cases where $G$ is a linear algebraic group or an Abelian variety. The computation is based on structure results on algebraic groups and Galois cohomology methods.
Categories: math.GR math.AG math.DG