  It is known observationally that the major axes of galaxy clusters and their
brightest cluster galaxies are roughly aligned with each other. To understand
the origin of the alignment, we identify 40 cluster-sized dark matter (DM)
haloes with masses higher than $5\times10^{13}~M_{\odot}$ and their central
galaxies (CGs) at $z\approx 0$ in the Horizon-AGN cosmological hydrodynamical
simulation. We trace the progenitors at 50 different epochs between $0<z<5$. We
then fit their shapes and orientations with a triaxial ellipsoid model. While
the orientations of both DM haloes and CGs change significantly due to repeated
mergers and mass accretions, their relative orientations are well aligned at
each epoch even at high redshifts, $z>1$. The alignment becomes tighter with
cosmic time; the major axes of the CGs and their host DM haloes at present are
aligned on average within $\sim 30^{\circ}$ in the three dimensional space and
$\sim 20^{\circ}$ in the projected plane. The orientations of the major axes of
DM haloes on average follow one of the eigen-vectors of the surrounding tidal
field that corresponds to the {\it slowest collapsing} (or even stretching)
mode, and the alignment with the tidal field also becomes tighter. This implies
that the orientations of CGs and DM haloes at the present epoch are largely
imprinted in the primordial density field of the Universe, whereas strong
dynamical interactions such as mergers are important to explain their mutual
alignment at each epoch.
