  In [1], Nakatsu and Takasaki have shown that the melting crystal model behind
the topological strings vertex provides a tau-function of the KP hierarchy
after an appropriate time deformation. We revisit their derivation with a focus
on the underlying quantum $W_{1+\infty}$ symmetry. Specifically, we point out
the role played by automorphisms and the connection with the intertwiner - or
vertex operator - of the algebra. This algebraic perspective allows us to
extend part of their derivation to the refined melting crystal model, lifting
the algebra to the quantum toroidal algebra of $\mathfrak{gl}(1)$ (also called
Ding-Iohara-Miki algebra). In this way, we take a first step toward the
definition of deformed hierarchies associated to A-model refined topological
strings.
