Beck modules over generalised Legendrian racks

A generalised Legendrian rack is a rack equipped with a Legendrian structure, which is a pair of self maps of the rack satisfying some axioms that are modelled on the Legendrian Reidemeister moves together with up and down cusps in the front diagram of an oriented Legendrian link. We prove that every generalised Legendrian rack admits a homogeneous representation in terms of cosets of a family of subgroups of a group. Further, using the concept of a trunk, which is analogous to a category, we define modules over generalised Legendrian racks. We also establish an equivalence between the category of modules over a fixed generalised Legendrian rack and the category of Beck modules over it.