  We study ultracold atoms in a finite size one-dimensional optical lattice
prepared in the Mott insulator phase and commonly coupled to a single cavity
mode. Due to resonance dipole-dipole interactions among the atoms, electronic
excitations delocalize and form {\it excitons}. These exciton modes are divided
into two groups: antisymmetric modes which decouple from the cavity mode
forming {\it dark states}, and symmetric modes significantly coupled to the
cavity mode called {\it bright states}. In typical setups the lowest and most
symmetric exciton is coupled to the cavity photons much stronger than the other
bright states and dominates the optical properties response of the atoms ({\it
superradiant state}). In the strong coupling regime this superradiant state is
coherently mixed with the cavity photon to form a doublet of polariton states
with the Rabi splitting.
