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On non-connected compact Lie groups. (Sur les groupes de Lie compacts non connexes.) (French) Zbl 0075.01602

Let \(G\) be a compact Lie group, \(G_0\) its identity component. In Chapter I the author studies the group \(A(G_0)\) of all automorphisms of \(G_0\), which is a Lie group whose identity component is the group \(I(G_0)\) of inner automorphisms of \(G_0\). It is shown that \(A(G_0)\) is a semi-direct product of \(I(G_0)\) and a certain discrete group \(U\). [An analogous theorem for semisimple Lie algebras is due to E. B. Dynkin, Dokl. Akad. Nauk SSSR, n. Ser. 76, 629–632 (1951; Zbl 0045.32103); G. D. Mostow, Mem. Am. Math. Soc. 14, 31–54 (1955; Zbl 0064.25901)]. If \(G_0\) is semi-simple, \(U\) is finite; it is determined explicitly for all simple compact groups. (On page 41, \(U\) is stated to be finite instead of discrete.) The group \(U\) is used to describe all group extensions \(E\) of simple groups \(G\), which are natural in the sense that the centralizer of \(G\), in \(E\) belongs to \(G_0\), and each automorphism of \(G_0\) is obtained by restriction of an inner automorphism of \(E\).
In Chapter II the author extends partially the theory of maximal tori for connected compact groups to non-connected compact groups \(G\). Let \(x\) be an element of \(G\), \(N_x\) the identity component of the normalizer of \(x\), \(T_0{}^h\) a maximal torus of \(N_x\), and \(T^{(h)}\) the closed subgroup of \(G\) generated by \(x\) and \(T_0{}^h\). Then \(T^{(h)}\) is the direct product of \(T_0{}^h\) and a finite cyclic group. If \(x\in G_0\), the dimension \(h=\text{rank}(G_0)\). If \(x\) varies through some other component \(G_1\) of \(G\), \(h\) is constant but in general \(h<\text{rank}(G_0)\). However, the various subgroups \(T^{(h)}\), \(x\in G_1\), are all conjugate under \(G_0\).
In Chapter III a diagram in the sense of Stiefel is introduced in each \(T^{(h)}\) and the analog of the Weyl group is defined. A procedure is given for the construction of the diagram of \(T^{(h)}\) from that of a maximal torus in \(G_0\) and this is carried out in detail in the case when \(G_0\) is simple. The result is applied to the extension \(A(G_0)\) of \(I(G_0)\), where \(G_0\) is simple, to describe the involutive automorphisms of simple compact Lie algebras.

MSC:

22Exx Lie groups
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