71 research outputs found
Twisted Weyl groups of Lie groups and nonabelian cohomology
For a cyclic group and a connected Lie group with an -module
structure (with the additional conditions that is compact and the
-module structure on is 1-semisimple if A\cong\ZZ), we define the
twisted Weyl group , which acts on and , where is
a maximal compact torus of , the identity component of the group of
invariants . We then prove that the natural map is a bijection, reducing the calculation of to the
calculation of the action of on . We also prove some properties of the
twisted Weyl group , one of which is that is a finite group. A new proof
of a known result concerning the ranks of groups of invariants with respect to
automorphisms of a compact Lie group is also given.Comment: 9 page
On the realization of Riemannian symmetric spaces in Lie groups II
In this paper we generalize a result in [1], showing that an arbitrary
Riemannian symmetric space can be realized as a closed submanifold of a
covering group of the Lie group defining the symmetric space. Some properties
of the subgroups of fixed points of involutions are also proved.Comment: 6 page
On the realization of Riemannian symmetric spaces in Lie groups
In this paper we give a realization of some symmetric space G/K as a closed
submanifold P of G. We also give several equivalent representations of the
submanifold P. Some properties of the set gK\cap P are also discussed, where gK
is a coset space in G.Comment: 8 page
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