The classification of global Lie wedges in sl(2)

Abstract

In [2] Hofmann and Hilgert investigate semigroups in SL(2, ℝ) and its universal covering group. Their interest is focussed on semigroups that are generated by the exponential image of a so-called Lie wedgeW in the Lie algebra. For an arbitrary subsemigroupS of a Lie groupG the set L(S) of subtangent vectors at the group identity1 is always a Lie wedge. IfS is closed, then$\overline {\left\langle {\exp L(S)} \right\rangle } \subseteq S$ is always true. If equality holds, thenS is called infinitesimally generated. Conversely, if we start with a Lie wedgeW, then it may happen that the inclusion$W \subseteq L\overline {\left\langle {\exp W} \right\rangle }$ is proper. Those wedges for which equality holds are called global. The classification of all global Lie wedges in a given Lie algebra is equivalent to classifying all infinitesimally generated subsemigroups of the corresponding simply connected Lie group. We give the solution to this classification problem for the Lie algebra sl(2, ℝ) . Roughly speaking, a Lie wedgeW ⊑sl(2, ℝ) is global if and only if it is either not pointed or it is contained in the intersection of two distinct halfspaces whose boundaries are subalgebras.