  We present a rigorous quantum scattering study of the effects of hyperfine
and Zeeman interactions on cold Li - H$_{2}$ collisions in the presence of an
external magnetic field using a recent ab initio potential energy surface. We
find that the low-field-seeking states of H$_{2}$ predominantly undergo elastic
collisions: the ratio of elastic-to-inelastic collisions exceeds 100 for
collision energies below 1.5 K. Furthermore, we demonstrate that most inelastic
collisions conserve the space-fixed projection of the nuclear spin. We show
that the anisotropic hyperfine interaction between the nuclear spin of H$_{2}$
and the electron spin of Li can have a significant effect on inelastic
scattering in the ultracold regime, as it mediates two processes: the electron
spin relaxation in lithium, and the nuclear spin - electron spin exchange.
Given the predominance of elastic collisions and the propensity of inelastic
collisions to retain H$_{2}$ in its low-field-seeking states, our results open
up the possibility of sympathetic cooling of molecular hydrogen by atomic
lithium, paving the way for future exploration of ultracold collisions and
high-precision spectroscopy of H$_{2}$ molecules.
