Globality in semisimple Lie groups

Karl-Hermann Neeb

Annales de l'institut Fourier (1990)

  • Volume: 40, Issue: 3, page 493-536
  • ISSN: 0373-0956

Abstract

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In the first section of this paper we give a characterization of those closed convex cones (wedges) W in the Lie algebra s l ( 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 S l ( 2 , R ) n ∼ , i.e., for which the subsemigroup S = ( exp W ) 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 W ⊆ L which are global in G , i.e., for which there exists a closed subsemigroup S ⊆ G having W as its tangent wedge L ( S ) .

How to cite

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Neeb, Karl-Hermann. "Globality in semisimple Lie groups." Annales de l'institut Fourier 40.3 (1990): 493-536. <http://eudml.org/doc/74886>.

@article{Neeb1990,
abstract = {In the first section of this paper we give a characterization of those closed convex cones (wedges) $W$ in the Lie algebra $sl(2,\{\bf R\})^n$ which are invariant under the maximal compact subgroup of the adjoint group and which are controllable in the associated simply connected Lie group $Sl(2,\{\bf R\})^\{n^\{\sim \}\}$, i.e., for which the subsemigroup $S=(\exp W)$ 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 $W\subseteq L$ which are global in $G$, i.e., for which there exists a closed subsemigroup $S\subseteq G$ having $W$ as its tangent wedge $L(S)$.},
author = {Neeb, Karl-Hermann},
journal = {Annales de l'institut Fourier},
keywords = {controllability; Lie semigroups; closed convex cones; wedges; root decompositions; invariant cones; semisimple Lie algebras; simply connected Lie group},
language = {eng},
number = {3},
pages = {493-536},
publisher = {Association des Annales de l'Institut Fourier},
title = {Globality in semisimple Lie groups},
url = {http://eudml.org/doc/74886},
volume = {40},
year = {1990},
}

TY - JOUR
AU - Neeb, Karl-Hermann
TI - Globality in semisimple Lie groups
JO - Annales de l'institut Fourier
PY - 1990
PB - Association des Annales de l'Institut Fourier
VL - 40
IS - 3
SP - 493
EP - 536
AB - In the first section of this paper we give a characterization of those closed convex cones (wedges) $W$ in the Lie algebra $sl(2,{\bf R})^n$ which are invariant under the maximal compact subgroup of the adjoint group and which are controllable in the associated simply connected Lie group $Sl(2,{\bf R})^{n^{\sim }}$, i.e., for which the subsemigroup $S=(\exp W)$ 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 $W\subseteq L$ which are global in $G$, i.e., for which there exists a closed subsemigroup $S\subseteq G$ having $W$ as its tangent wedge $L(S)$.
LA - eng
KW - controllability; Lie semigroups; closed convex cones; wedges; root decompositions; invariant cones; semisimple Lie algebras; simply connected Lie group
UR - http://eudml.org/doc/74886
ER -

References

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