Remarks on the structure and integrability of LA-groups

We study the structure of an LA-group identifying its underlying VB-group with a representation up to homotopy. We show that the Lie algebroid structure is determined by a complementary action up to homotopy of the Lie algebra of units. We identify the equations that the representation and the action need to verify in order to assemble into an LA-group, establishing an equivalence between LA-groups and LA-matched pairs. As an application, we catalog some extreme examples of representations and actions and comment on their integrability.