Liouville reflection operator, affine Yangian and Bethe ansatz

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.

Fractional integration operator on some radial rays and intertwining for the Dunkl operator

In this paper we consider the differential-difference reflection operator associated with a finite cyclic group,It is to emphasize that both hyper–Bessel operators and the so-called Poisson–Dimovski transformation (transmutation) are typical examples of the operators of generalized fractional calculus [