Bethe subalgebras in affine Birman–Murakami–Wenzl algebras and flat connections for q-KZ equations

Abstract

Commutative sets of Jucys–Murphy elements for affine braid groups of ${A}^{(1)},{B}^{(1)},{C}^{(1)},{D}^{(1)}$ types were defined. Construction of R-matrix representations of the affine braid group of type ${C}^{(1)}$ and its distinguished commutative subgroup generated by the ${C}^{(1)}$-type Jucys–Murphy elements are given. We describe a general method to produce flat connections for the two-boundary quantum Knizhnik–Zamolodchikov equations as necessary conditions for Sklyanin's type transfer matrix associated with the two-boundary multicomponent Zamolodchikov algebra to be invariant under the action of the ${C}^{(1)}$-type Jucys–Murphy elements. We specify our general construction to the case of the Birman–Murakami–Wenzl algebras (BMW algebras for short). As an application we suggest a baxterization of the Dunkl–Cherednik elements ${Y}^{\prime }{\rm{s}}$ in the double affine Hecke algebra of type A.