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

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).

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.

@article{AIF_1990__40_3_493_0,
     author = {Neeb, Karl-Hermann},
     title = {Globality in semisimple Lie groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {40},
     number = {3},
     year = {1990},
     pages = {493-536},
     doi = {10.5802/aif.1222},
     zbl = {0703.17003},
     mrnumber = {92h:17005},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1990__40_3_493_0}
}
Neeb, Karl-Hermann. Globality in semisimple Lie groups. Annales de l'Institut Fourier, Volume 40 (1990) no. 3, pp. 493-536. doi : 10.5802/aif.1222. http://www.numdam.org/item/AIF_1990__40_3_493_0/

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