Invariant orders in Lie groups

Neeb, Karl-Hermann

  • Proceedings of the Winter School "Geometry and Physics", Publisher: Circolo Matematico di Palermo(Palermo), page [217]-221

Abstract

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[For the entire collection see Zbl 0742.00067.]The author formulates several theorems about invariant orders in Lie groups (without proofs). The main theorem: a simply connected Lie group G admits a continuous invariant order if and only if its Lie algebra L ( G ) contains a pointed invariant cone. V. M. Gichev has proved this theorem for solvable simply connected Lie groups (1989). If G is solvable and simply connected then all pointed invariant cones W in L ( G ) are global in G (a Lie wedge W ⊂ L ( G ) is said to be global in G if W = L ( S ) for a Lie semigroup S ⊂ G ). This is false in general if G is a simple simply connected Lie group.

How to cite

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Neeb, Karl-Hermann. "Invariant orders in Lie groups." Proceedings of the Winter School "Geometry and Physics". Palermo: Circolo Matematico di Palermo, 1991. [217]-221. <http://eudml.org/doc/219983>.

@inProceedings{Neeb1991,
abstract = {[For the entire collection see Zbl 0742.00067.]The author formulates several theorems about invariant orders in Lie groups (without proofs). The main theorem: a simply connected Lie group $G$ admits a continuous invariant order if and only if its Lie algebra $L(G)$ contains a pointed invariant cone. V. M. Gichev has proved this theorem for solvable simply connected Lie groups (1989). If $G$ is solvable and simply connected then all pointed invariant cones $W$ in $L(G)$ are global in $G$ (a Lie wedge $W\subset L(G)$ is said to be global in $G$ if $W=L(S)$ for a Lie semigroup $S\subset G$). This is false in general if $G$ is a simple simply connected Lie group.},
author = {Neeb, Karl-Hermann},
booktitle = {Proceedings of the Winter School "Geometry and Physics"},
keywords = {Srni (Czechoslovakia); Proceedings; Winter school; Geometry; Physics},
location = {Palermo},
pages = {[217]-221},
publisher = {Circolo Matematico di Palermo},
title = {Invariant orders in Lie groups},
url = {http://eudml.org/doc/219983},
year = {1991},
}

TY - CLSWK
AU - Neeb, Karl-Hermann
TI - Invariant orders in Lie groups
T2 - Proceedings of the Winter School "Geometry and Physics"
PY - 1991
CY - Palermo
PB - Circolo Matematico di Palermo
SP - [217]
EP - 221
AB - [For the entire collection see Zbl 0742.00067.]The author formulates several theorems about invariant orders in Lie groups (without proofs). The main theorem: a simply connected Lie group $G$ admits a continuous invariant order if and only if its Lie algebra $L(G)$ contains a pointed invariant cone. V. M. Gichev has proved this theorem for solvable simply connected Lie groups (1989). If $G$ is solvable and simply connected then all pointed invariant cones $W$ in $L(G)$ are global in $G$ (a Lie wedge $W\subset L(G)$ is said to be global in $G$ if $W=L(S)$ for a Lie semigroup $S\subset G$). This is false in general if $G$ is a simple simply connected Lie group.
KW - Srni (Czechoslovakia); Proceedings; Winter school; Geometry; Physics
UR - http://eudml.org/doc/219983
ER -

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