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

TL;DR

This paper constructs flat connections for q-KZ equations using Bethe subalgebras in affine BMW algebras, introducing new methods for invariance conditions and baxterization of Dunkl-Cherednik elements.

Contribution

It develops a general method to produce flat connections for q-KZ equations via Bethe subalgebras in affine BMW algebras, extending the theory of Jucys-Murphy elements.

Findings

Defined commutative Jucys-Murphy elements for affine braid groups.

Constructed R-matrix representations and identified a commutative subgroup.

Proposed a baxterization of Dunkl-Cherednik elements in type A.

Abstract

Commutative sets of Jucys-Murphyelements 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 distinguish 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. As an application we suggest a baxterization of the Dunkl--Cherednik elements $Y's$ in the double affine Hecke algebra of type $A$.