Semigroups in the simply connected covering of \(SL(2)\)
This article contributes to the investigation of subsemigroups of the simply connected covering group \(G=\text{Sl}(2,\mathbb{R})\tilde{ }\) of \(\text{Sl}(2,\mathbb{R})\) which are Lie semigroups, i.e., closed and topologically generated by one-parameter semigroups. In the Lie algebra \({\mathfrak {sl}}(2)\) there exists a one-parameter family \(W_ r\), \(r>0\) of Lorentzian cones which are invariant under the adjoint action of SO(2). In [Semigroup Forum 43, No. 1, 33-43 (1991; Zbl 0736.22005)] the reviewer proved that \(\exp W_ r\) generates the whole group \(G\) if and only if \(r>1\). Later \textit{J. Hilgert} showed [in Math. Z. 209, 463-466 (1992)] that these results can be derived from rather general arguments which are derived from those ideas which led Gödel to his non-causal cosmological model. In this article the author uses the Pontrjagin Maximum Principle to exhibit smooth causal loops for \(r>1\) (the non-causal case) and he also derives an explicit description of the semigroups \(S_ r:=\langle\exp W_ r\rangle\) for \(r\leq 1\). His method also leads to a more explicit description of the exponential function of \(G\).
scientific article; zbMATH DE number 164975