  Monolayer (ML) semiconducting transition-metal dichalcogenides (TMDs)
represent a unique class of two-dimensional (2D) electron systems. Their
atomically thin structure -- just like graphene -- facilitates gate-tunability,
while the sizable band gap and strong spin-orbit coupling hold promise for
properties beyond graphene. Measurements under large magnetic fields have
revealed an unusual LL structure, distinct from other 2D electron systems.
However, owing to limited sample quality and poor electrical contact, probing
the lowest Landau levels (LLs) has been challenging, and observation of
electron correlations within the fractionally filled LLs regime has not been
possible. Here, through bulk electronic compressibility measurements, we
investigate the LL structure of ML WSe$_2$ in the extreme quantum limit, and
observe fractional quantum Hall (FQH) states in the lowest three LLs. The
odd-denominator FQH sequences demonstrate a systematic evolution with the LL
orbital index, which has not been observed in any other system but is
consistent with generic theoretical expectations. In addition, we observe an
even-denominator state in the second LL that is expected to host non-Abelian
statistics. Our results suggest that the 2D semiconductors can provide an
experimental platform that closely resembles idealized theoretical models in
the quantum Hall regime.
