Fragmentation at height associated with Lévy processes

We consider the height process of a Lévy process with no negative jumps, and its associated continuous tree representation. Using tools developed by Duquesne and Le Gall, we construct a fragmentation process at height, which generalizes the fragmentation at height of stable trees given by Miermont. In this more general framework, we recover that the dislocation measures are the same as the dislocation measures of the fragmentation at nodes introduced by Abraham and Delmas, up to a factor equal to the fragment size. We also compute the asymptotics for the number of small fragments.