  We show how coupling to an Einstein phonon $\omega_E$ affects the absorption
peaks seen in the optical conductivity of graphene under a magnetic field $B$.
The energies and widths of the various lines are shifted, and additional peaks
arise in the spectrum. Some of these peaks are Holstein sidebands, resulting
from the transfer of spectral weight in each Landau level (LL) into
phonon-assisted peaks in the spectral function. Other additional absorption
peaks result from transitions involving split LLs, which occur when a LL falls
sufficiently close to a peak in the self-energy. We establish the selection
rules for the additional transitions and characterize the additional absorption
peaks. For finite chemical potential, spectral weight is asymmetrically
distributed about the Dirac point; we discuss how this causes an asymmetry in
the transitions due to left- and right-handed circularly polarized light and
therefore oscillatory behavior in the imaginary part of the off-diagonal Hall
conductivity. We also find that the semiclassical cyclotron resonance region is
renormalized by an effective-mass factor but is not directly affected by the
additional transitions. Last, we discuss how the additional transitions can
manifest in broadened, rather than split, absorption peaks due to large
scattering rates seen in experiment.
