  The efficiency of optical trapping of ultracold atoms depend on the atomic
dynamic dipole polarizability governing the atom-field interaction. In this
article, we have calculated the real and imaginary parts of the dynamic dipole
polarizability of dysprosium in the ground and first excited level. Due to the
high electronic angular momentum of those two states, the polarizabilities
possess scalar, vector and tensor contributions that we have computed, on a
wide range of trapping wavelengths, using the sum-over-state formula. Using the
same formalism, we have also calculated the $C_6$ coefficients characterizing
the van der Waals interaction between two dysprosium atoms in the two lowest
levels. We have computed the energies of excited states and the transition
probabilities appearing in the sums, using a combination of \textit{ab initio}
and least-square-fitting techniques provided by the Cowan codes and extended in
our group. Regarding the real part of the polarizability, for field frequencies
far from atomic resonances, the vector and tensor contributions are
two-order-of-magnitude smaller than the scalar contribution, whereas for the
imaginary part, the vector and tensor contributions represent a noticeable
fraction of the scalar contribution. This offers the possibility to control the
decoherence and trap losses due to spontaneous emission.
