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Abstract

A Lorentzian coneW in a finite dimensional real Lie algebraL is the convex closed cone bounded by one half of the zero-set of a Lorentzian formq onL with the additional property, that for all sufficiently smallx, yW the Campbell-Hausdorff productx*y=x+y+1/2[x,y]+..., is also inW. We characterize Lorentzian cones completely; in particular, with the exception of one class of “almost abelian” solvable algebras, the Lorentzian formq is invariant, i.e., satisfiesq([x, y], z)=q (x,[y, z]).

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Hilgert, J., Hofmann, K.H. Lorentzian cones in real Lie algebras. Monatshefte für Mathematik 100, 183–210 (1985). https://doi.org/10.1007/BF01299267

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