  We study electronic excitations of a degenerate gas of atoms trapped in pairs
in an optical lattice. Local dipole-dipole interactions produce a long lived
antisymmetric and a short lived symmetric superposition of individual atomic
excitations as the lowest internal on-site excitations. Due to the much larger
dipole moment the symmetric states couple efficiently to neighbouring lattice
sites and can be well represented by Frenkel excitons, while the antisymmetric
dark states stay localized. Within a cavity only symmetric states couple to
cavity photons inducing long range interactions to form polaritons. We
calculate their dispersion curves as well as cavity transmission and reflection
spectra to observe them. For a lattice with aspherical sites bright and dark
states get mixed and their relative excitation energies depend on photon
polarizations. The system should allow to study new types of solid state
phenomena in atom filled optical lattices.
