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

A note on the algebraic engineering of 4D $\mathcal{N}=2$ super Yang-Mills theories