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Invariant orders in Lie groups. (English) Zbl 0755.22003
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:
 2.2e+16 General properties and structure of real Lie groups