54 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
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
Zero patterns and unitary similarity
A subspace of the space, L(n), of traceless complex matrices can
be specified by requiring that the entries at some positions be zero.
The set, , of these positions is a (zero) pattern and the corresponding
subspace of L(n) is denoted by . A pattern is universal if every
matrix in L(n) is unitarily similar to some matrix in . The problem of
describing the universal patterns is raised, solved in full for , and
partial results obtained for . Two infinite families of universal patterns
are constructed. They give two analogues of Schur's triangularization theorem.Comment: 39 page
- β¦