  We study six luminous Lyman-alpha emitters (LAEs) with very blue rest-frame
UV continua at $5.7\le z \le 6.6$. These LAEs have previous HST and Spitzer
IRAC observations. Combining our newly acquired HST images, we find that their
UV-continuum slopes $\beta$ are in a range of $-3.4\le \beta \le -2.6$. Unlike
previous, tentative detections of $\beta \simeq -3$ in photometrically
selected, low-luminosity galaxies, our LAEs are spectroscopically confirmed and
luminous ($M_{\rm UV}<-20$ mag). We model their broadband spectral energy
distributions (SEDs), and find that two $\beta\simeq-2.6\pm0.2$ galaxies can be
well fitted with young and dust-free stellar populations. However, it becomes
increasingly difficult to fit bluer galaxies. We explore further
interpretations by including non-zero LyC escape fraction $f_{\rm esc}$, very
low metallicities, and/or AGN contributions. Assuming $f_{\rm esc}\simeq0.2$,
we achieve the bluest slopes $\beta\simeq-2.7$ when nebular emission is
considered. This can nearly explain the SEDs of two galaxies with
$\beta\simeq-2.8$ and --2.9 ($\sigma_{\beta}=0.15$). Larger $f_{\rm esc}$
values and very low metallicities are not favored by the strong nebular line
emission (evidenced by the IRAC flux) or the observed (IRAC 1 - IRAC 2) color.
Finally, we find that the $\beta\simeq-2.9$ galaxy can potentially be well
explained by the combination of a very young population with a high $f_{\rm
esc}$ ($\ge0.5$) and an old, dusty population. We are not able to produce two
$\beta \simeq -3.4 \pm0.4$ galaxies. Future deep spectroscopic observations are
needed to fully understand these galaxies.
