  While various observations measured ellipticities of galaxy clusters and
alignments between orientations of the brightest cluster galaxies and their
host clusters, there are only a handful of numerical simulations that implement
realistic baryon physics to allow direct comparisons with those observations.
Here we investigate ellipticities of galaxy clusters and alignments between
various components of them and the central galaxies in the state-of-the-art
cosmological hydrodynamical simulation Horizon-AGN, which contains dark matter,
stellar, and gas components in a large simulation box of $(100 h^{-1}$ Mpc$)^3$
with high spatial resolution ($\sim1$ kpc). We estimate ellipticities of total
matter, dark matter, stellar, gas surface mass density distributions, X-ray
surface brightness, and the Compton $y$-parameter of the Sunyaev-Zel'dovich
effect, as well as alignments between these components and the central galaxies
for 120 projected images of galaxy clusters with masses
$M_{200}>5\times10^{13}M_{\odot}$. Our results indicate that the distributions
of these components are well aligned with the major-axes of the central
galaxies, with the root mean square value of differences of their position
angles of $\sim 20^\circ$, which vary little from inner to the outer regions.
We also estimate alignments of these various components with total matter
distributions, and find tighter alignments than those for central galaxies with
the root mean square value of $\sim 15^\circ$. We compare our results with
previous observations of ellipticities and position angle alignments and find
reasonable agreements. The comprehensive analysis presented in this paper
provides useful prior information for analyzing stacked lensing signals as well
as designing future observations to study ellipticities and alignments of
galaxy clusters.
