  Some BPS quantities of $\mathcal{N}=1$ 5D quiver gauge theories, like
instanton partition functions or qq-characters, can be constructed as algebraic
objects of the Ding-Iohara-Miki (DIM) algebra. This construction is applied
here to $\mathcal{N}=2$ super Yang-Mills theories in four dimensions using a
degenerate version of the DIM algebra. We build up the equivalent of horizontal
and vertical representations, the first one being defined using vertex
operators acting on a free boson's Fock space, while the second one is
essentially equivalent to the action of Vasserot-Shiffmann's Spherical Hecke
central algebra. Using intertwiners, the algebraic equivalent of the
topological vertex, we construct a set of $\mathcal{T}$-operators acting on the
tensor product of horizontal modules, and the vacuum expectation values of
which reproduce the instanton partition functions of linear quivers. Analysing
the action of the degenerate DIM algebra on the $\mathcal{T}$-operator in the
case of a pure $U(m)$ gauge theory, we further identify the degenerate version
of Kimura-Pestun's quiver W-algebra as a certain limit of q-Virasoro algebra.
Remarkably, as previously noticed by Lukyanov, this particular limit reproduces
the Zamolodchikov-Faddeev algebra of the sine-Gordon model.
