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