## Abstract

Connections between set-theoretic Yang-Baxter and reflection

equations and quantum integrable systems are investigated. We show that set-theoretic R-matrices are expressed as twists of known solutions. We then focus on reflection and twisted algebras and we derive the associated defining algebra relations for R-matrices being Baxterized solutions of the A-type Hecke algebra H

proof of the duality between the boundary finite subalgebra and the B-type Hecke algebra. These are universal statements that largely generalize previous relevant findings, and also allow the investigation of the symmetries of the double row transfer matrix.

equations and quantum integrable systems are investigated. We show that set-theoretic R-matrices are expressed as twists of known solutions. We then focus on reflection and twisted algebras and we derive the associated defining algebra relations for R-matrices being Baxterized solutions of the A-type Hecke algebra H

_{N}(q = 1). We show in the case of the reflection algebra that there exists a “boundary” finite sub-algebra for some special choice of “boundary” elements of the B-type Hecke algebra B_{N}(q = 1, Q). We also show the key proposition that the associated double row transfer matrix is essentially expressed in terms of the elements of the B-type Hecke algebra. This is one of the fundamental results of this investigation together with theproof of the duality between the boundary finite subalgebra and the B-type Hecke algebra. These are universal statements that largely generalize previous relevant findings, and also allow the investigation of the symmetries of the double row transfer matrix.

Original language | English |
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Article number | 105 |

Journal | Letters in Mathematical Physics |

Volume | 111 |

Issue number | 4 |

Early online date | 2 Aug 2021 |

DOIs | |

Publication status | Published - Aug 2021 |

## Keywords

- Braces
- Hecke algebras
- Open quantum spin chains
- Reflection equation
- Set-theoretic Yang-Baxter equation

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics