In this paper we construct a new family of representations for the quantum toroidal algebra
, which are
-extremal
in the sense of Hernandez. We construct extremal loop weight modules associated to
level fundamental
weights
when
is
odd and
,
or .
To do this, we relate monomial realizations of level 0 extremal
fundamental weight crystals to integrable representations of
, and
we introduce promotion operators for the level 0 extremal fundamental weight
crystals. By specializing the quantum parameter, we get finite-dimensional modules
of quantum toroidal algebras at roots of unity. In general, we give a conjectural
process to construct extremal loop weight modules from monomial realizations of
crystals.
