Chern-Simons and palatini actions and (2+1)-gravity

In this letter we make three observations: (i) Given any Lie group, G, the (2+1)-Palatini action is equivalent to the Chern-Simons action for an "inhomogenized" Lie group, IG, constructed from G. (ii) Since the reduced phase space of the Chern-Simons theory based on any group G is the reduced configuration space of the Palatini theory based on the same G - i.e., the Chern-Simons theory for IG-one can introduce additional structure on the configuration and phase-spaces of Palatini theories -including (2+1)-gravity-which plays an important role in their quantization. (iii) Using the first observation, one can couple Chern-Simons matter to (2+1)-gravity in a straightforward way. We point out that there is, however, a qualitative difference between this coupling and that of matter with genuine, local degrees of freedom.