Fundamental groups of semisimple symmetric spaces. II
Let \({\mathfrak g}\) be a real simple Lie algebra. The universal linear group \(G_{ul}\) of \({\mathfrak g}\) is defined as the analytic subgroup corresponding to \({\mathfrak g}\) of the simple connected complex Lie group whose Lie algebra is the complexification of \({\mathfrak g}\). Let \(\sigma\) be an involution of \({\mathfrak g}\). Denote by \({\mathfrak g}^\sigma\) (resp., \(G^\sigma_{ul})\) the \(\sigma\)-invariant subalgebra of \({\mathfrak g}\) (resp., subgroup of \(G_{ul}) \). Then using the classification of the restricted root system of the symmetric pair \(({\mathfrak g}, {\mathfrak g}^\sigma)\), the author proves that, for a simple Lie algebra \({\mathfrak g}\) of exceptional type, the fundamental group \(\pi_1(G_{ul}/G_{ul}^\sigma)\) of the semisimple symmetric space \(G_{ul}/G^\sigma_{ul}\) is isomorphic to 1, \(\mathbb{Z}_2\) or \(\mathbb{Z}\). The details of calculation are summarized in four tables. In Part I of this paper [Representation of Lie groups, Adv. Stud. Pure Math. 14, 519-529 (1988; Zbl 0723.22021)] the fundamental group \(\pi_1 (G/G^\sigma)\) was determined for the adjoint group \(G\) of \({\mathfrak g}\). In almost all cases the fundamental group \(\pi_1(G/G^\sigma)\) coincides with \(\pi_1 (G_{ul}/G^\sigma_{ul})\). The cases where these two groups are different are listed in Table 2.
scientific article; zbMATH DE number 989613