From affine Hecke algebras to boundary symmetries

TL;DR

This paper explores the connection between affine Hecke algebras and boundary symmetries in open spin chains, deriving non-diagonal solutions to the reflection equation and analyzing their symmetry properties.

Contribution

It introduces new realizations of the affine Hecke algebra to recover known solutions and constructs boundary non-local charges that reveal symmetries of open spin chains.

Findings

Recovered non-diagonal solutions of the reflection equation for $U_q(\ ext{hat} gl_n)$

Constructed boundary non-local charges commuting with the Hamiltonian

Identified boundary conditions under which transfer matrices commute with non-local charges

Abstract

Motivated by earlier works we employ appropriate realizations of the affine Hecke algebra and we recover previously known non-diagonal solutions of the reflection equation for the $U_{q}(\hat{gl_n})$ case. The corresponding $N$ site spin chain with open boundary conditions is then constructed and boundary non-local charges associated to the non-diagonal solutions of the reflection equation are derived, as coproduct realizations of the reflection algebra. With the help of linear intertwining relations involving the aforementioned solutions of the reflection equation, the symmetry of the open spin chain with the corresponding boundary conditions is exhibited, being essentially a remnant of the $U_{q}(\hat{gl_{n}})$ algebra. More specifically, we show that representations of certain boundary non-local charges commute with the generators of the affine Hecke algebra and with the local Hamiltonian of the open spin chain for a particular choice of boundary conditions. Furthermore, we are able to show that the transfer matrix of the open spin chain commutes with a certain number of boundary non-local charges, depending on the choice of boundary conditions.