In these notes we study integrable structure of conformal field theory by means of Liouville reflection operator/Maulik Okounkov R-matrix. We discuss relation between RLL and current realization of the affine Yangian of $$ \mathfrak{gl}(1) $$ gl 1 . We construct the family of commuting transfer matrices related to the Intermediate Long Wave hierarchy and derive Bethe ansatz equations for their spectra discovered by Nekrasov and Okounkov and independently by one of the authors. Our derivation mostly follows the one by Feigin, Jimbo, Miwa and Mukhin, but is adapted to the conformal case.

The deformed W algebras of type A have a uniform description in terms of the quantum toroidal gl 1 algebra E. We introduce a comodule algebra K over E which gives a uniform construction of basic deformed W currents and screening operators in types B, C, D including twisted and supersymmetric cases. We show that a completion of algebra K contains three commutative subalgebras. In particular, it allows us to obtain a commutative family of integrals of motion associated with affine Dynkin diagrams of all non-exceptional types except D(2) ℓ+1 . We also obtain in a uniform way deformed finite and affine Cartan matrices in all classical types together with a number of new examples, and discuss the corresponding screening operators.

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