  We investigate theoretically the interaction of polar molecules with optical
lattices and microwave fields. We demonstrate the existence of frequency
windows in the optical domain where the complex internal structure of the
molecule does not influence the trapping potential of the lattice. In such
frequency windows the Franck-Condon factors are so small that near-resonant
interaction of vibrational levels of the molecule with the lattice fields have
a negligible contribution to the polarizability and light-induced decoherences
are kept to a minimum. In addition, we show that microwave fields can induce a
tunable dipole-dipole interaction between ground-state rotationally symmetric
(J=0) molecules. A combination of a carefully chosen lattice frequency and
microwave-controlled interaction between molecules will enable trapping of
polar molecules in a lattice and possibly realize molecular quantum logic
gates. Our results are based on ab initio relativistic electronic structure
calculations of the polar KRb and RbCs molecules combined with calculations of
their rovibrational motion.
