  Poisson actions of Poisson Lie groups have an interesting and rich geometric
structure. We will generalize some of this structure to Dirac actions of Dirac
Lie groups. Among other things, we extend a result of Jiang-Hua-Lu, which
states that the cotangent Lie algebroid and the action algebroid for a Poisson
action form a matched pair. We also give a full classification of Dirac actions
for which the base manifold is a homogeneous space $H/K$, obtaining a
generalization of Drinfeld's classification for the Poisson Lie group case.
