Main BPS objects of Super Yang-Mills theories with enough supersymmetry can be computed exactly by localization (i.e. 4d N=2, 5d N=1, 6d N=(2,0)). They exhibit a form of integrability expressed as covariance under Yangian/Toroidal algebras. These algebras are formally equivalent to (quantum/elliptic) $W_{1+infty}$, which implies the AGT-correspondence with (q-deformed/elliptic) conformal blocks. par Here we will focus on Nekrasov instanton partition functions for linear quivers in N=1 5d SYM. These partition functions can be written using elements of the representation theory of (Ding-Iohara-Miki) quantum toroidal $gl_1$ algebra: coherent states, intertwiners, vertex operators,… As a result, Schwinger-Dyson identities take the form of a regularity property for a resolvant, the qq-character. We will also comment on similar results obtained for 4d N=2 theories in the degenerate limit, and expected generalization to 6d theories.

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