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Invariant Lorentzian orders on simply connected Lie groups. (English) Zbl 0652.22008

The study of the causal structure of space-time has led a number of authors to consider partially ordered Lie groups. With an invariant semigroup of positive elements, there is associated an invariant positive cone in the Lie algebra. The author considers the case of an invariant Lorentzian cone (one half of the double cone of a Lorentzian quadratic form). It is shown that these only arise as the direct sum of a compact Lie algebra equipped with a positive definite form and either the reals (with a negative form), sl(2,R) with its Killing form, or one of the solvable algebras called oscillator algebras equipped with an invariant Lorentzian form. From this result it follows that if G is a corresponding simply connected Lie group, there exists a directed, infinitesimally generated, invariant partial order on G such that the given cone is the tangent object of the set of positive elements.
Reviewer: J.D.Lawson

MSC:

22E15 General properties and structure of real Lie groups
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
06F25 Ordered rings, algebras, modules
83C99 General relativity
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