Invariant orders in Lie groups. (English) Zbl 0755.22003

Neeb, Karl-Hermann

Geometry and physics, Proc. Winter Sch., Srni/Czech. 1990, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 26, 217-221 (1991).

[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.

Reviewer: A.K.Guts (Omsk)

MSC:

22E15

General properties and structure of real Lie groups

Keywords:

invariant orders in Lie groups; simply connected Lie group; Lie algebra; pointed invariant cones; Lie wedge; Lie semigroup

Citations:

Zbl 0742.00067