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arXiv · 1510.05374

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

A. P. Isaev·A. N. Kirillov·V. O. Tarasov
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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 $A$.

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A. P. Isaev, A. N. Kirillov, V. O. Tarasov. 2016-02-17. Bethe subalgebras in affine Birman--Murakami--Wenzl algebras and flat connections for q-KZ equations. https://doi.org/10.1088/1751-8113%2F49%2F20%2F204002

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