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Type: Artigo de periódico
Abstract: Let G be a connected semi-simple Lie group with finite center and S subset-of G a subsemigroup with interior points. Let G/L be a homogeneous space. There is a natural action of S on G/L. The relation x less-than-or-equal-to y if y is-an-element-of Sx, x,y is-an-element-of G/L, is transitive but not reflexive nor symmetric. Roughly, a control set is a subset D subset-of G/L, inside of which reflexivity and symmetry for less-than-or-equal-to hold. Control sets are studied in G/L when L is the minimal parabolic subgroup. They are characterized by means of the Weyl chambers in G meeting int S. Thus, for each w is-an-element-of W, the Weyl group of G, there is a control set of D(w). D1 is the only invariant control set, and the subset W(S) = {w : D(w) = D1} turns out to be a subgroup. The control sets are determined by W(S)/W. The following consequences are derived: i) S = G if S is transitive on G/H, i.e. Sx = G/H for all x is-an-element-of G/H. Here H is a non discrete closed subgroup different from G and G is simple. ii) S is neither left nor right reversible if S not-equal G. iii) S is maximal only if it is the semigroup of compressions of a subset of some minimal flag manifold.
Editor: Springer Verlag
Rights: fechado
Identifier DOI: 10.1007/BF02573505
Date Issue: 1995
Appears in Collections:Artigos e Materiais de Revistas Científicas - Unicamp

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