MacMahon KZ equation for Ding-Iohara-Miki algebra

We derive a generalized Knizhnik-Zamolodchikov equation for the correlation function of the intertwiners of the vector and the MacMahon representations of Ding-Iohara-Miki algebra. These intertwiners are cousins of the refined topological vertex which is regarded as the intertwining operator of the Fock representation. The shift of the spectral parameter of the intertwiners is generated by the operator which is constructed from the universal R matrix. The solutions to the generalized KZ equation are factorized into the ratio of two point functions which are identified with generalizations of the Nekrasov factor for supersymmetric quiver gauge theories.

KZ equations and Bethe subalgebras in generalized Yangians related to compatible $R$-matrices

The notion of compatible braidings was introduced in Isaev et al. (1999, J. Phys. A, 32, L115–L121). On the base of this notion, the authors of Isaev et al. (1999, J. Phys. A, 32, L115–L121) defined certain quantum matrix algebras generalizing the RTT algebras and Reflection Equation ones. They also defined analogues of some symmetric polynomials in these algebras and showed that these polynomials generate commutative subalgebras, called Bethe. By using a similar approach, we introduce certain new algebras called generalized Yangians and define analogues of some symmetric polynomials in these algebras. We claim that they commute with each other and thus generate a commutative Bethe subalgebra in each generalized Yangian. Besides, we define some analogues (also arising from couples of compatible braidings) of the Knizhnik–Zamolodchikov equation—classical and quantum. Communicated by: Alexander Veselov

Integrable Stochastic Dualities and the Deformed Knizhnik–Zamolodchikov Equation

We present a new method for obtaining duality functions in multi-species asymmetric exclusion processes (mASEP), from solutions of the deformed Knizhnik–Zamolodchikov (KZ) equations. Our method reproduces, as a special case, duality functions for the self-dual single species ASEP on the integer lattice.