Globality in semisimple Lie groups
Annales de l'Institut Fourier, Tome 40 (1990) no. 3, pp. 493-536.

Dans la première section de cet article nous caractérisons les cônes convexes fermés de W de l’algèbre de Lie sl(2,R)n, qui sont invariants sous l’action d’un groupe compact maximal du groupe adjoint et qui sont contrôlables dans le groupe Sl(2,R)n, c’est-à-dire tels que l’image exponentielle de W engendre le groupe tout entier (Theorem 1.3). Dans la section 2 nous développons des instruments algébriques concernant le système de racines réelles relatives à une sous-algèbre de Cartan compacte plongée et les cônes invariants dans les algèbres de Lie semi-simples. Dans la section 3 nous utilisons ces instruments, en combinaison avec des résultats de la section 1, pour caractériser les cônes invariants dans une algèbres de Lie semi-simples qui sont contrôlables dans le groupe simplement connexe associé. Si L est simple nous obtenons une caractérisation des cônes invariants WL qui sont globaux, c’est-à-dire pour lesquels il existe un semi-groupe fermé SG avec L(S)=W.

In the first section of this paper we give a characterization of those closed convex cones (wedges) W in the Lie algebra sl(2,R)n which are invariant under the maximal compact subgroup of the adjoint group and which are controllable in the associated simply connected Lie group Sl(2,R)n, i.e., for which the subsemigroup S=(expW) generated by the exponential image of W agrees with the whole group G (Theorem 13). In Section 2 we develop some algebraic tools concerning real root decompositions with respect to compactly embedded Cartan algebras and invariant cones in semisimple Lie algebras. In Section 3 these tools, combined with the results from Section 1, yield a characterization of those invariant cones in a semisimple Lie algebra L which are controllable in the associated simply connected Lie group G. If L is simple, we even get a characterization of those invariant wedges WL which are global in G, i.e., for which there exists a closed subsemigroup SG having W as its tangent wedge L(S).

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     author = {Neeb, Karl-Hermann},
     title = {Globality in semisimple {Lie} groups},
     journal = {Annales de l'Institut Fourier},
     pages = {493--536},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {40},
     number = {3},
     year = {1990},
     doi = {10.5802/aif.1222},
     mrnumber = {92h:17005},
     zbl = {0703.17003},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1222/}
}
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Neeb, Karl-Hermann. Globality in semisimple Lie groups. Annales de l'Institut Fourier, Tome 40 (1990) no. 3, pp. 493-536. doi : 10.5802/aif.1222. https://www.numdam.org/articles/10.5802/aif.1222/

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