  In this paper we construct a new family of representations for the quantum
toroidal algebras of type $A_n$, which are $\ell$-extremal in the sense of
Hernandez [24]. We construct extremal loop weight modules associated to level 0
fundamental weights $\varpi_\ell$ when $n=2r+1$ is odd and $\ell=1, r+1$ or
$n$. To do it, we relate monomial realizations of level 0 extremal fundamental
weight crystals with integrable representations of
$\mathcal{U}_q(sl_{n+1}^{tor})$, 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.
