  Ce$_{3}$Al is an archetypal heavy-fermion compound with multiple crystalline
phases. Here, we try to investigate its electronic structures in the hexagonal
phase ($\alpha$-Ce$_{3}$Al) and cubic phase ($\beta$-Ce$_{3}$Al) by means of a
combination of density functional theory and single-site dynamical mean-field
theory. We confirm that the 4$f$ valence electrons in both phases are
itinerant, accompanied with strong valence state fluctuations. Their 4$f$ band
structures are heavily renormalized by electronic correlations, resulting in
large effective electron masses. The Kondo screening in Ce$_{3}$Al would be
protracted over a wide range of temperature since the single-impurity Kondo
temperature $T_{K}$ is much higher than the coherent Kondo temperature
$T^{*}_{K}$. Especially, the crystal structure of $\alpha$-Ce$_{3}$Al forms a
layered kagome lattice. We observe conspicuous kagome-derived flat bands and
Dirac cones (or gaps) in its quasiparticle band structure. Therefore, it is
concluded that the hexagonal phase of Ce$_{3}$Al will be a promising candidate
of heavy-fermion kagome metal.
