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 {mathcal {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 {mathcal {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 the 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.
Highlights
The Yang–Baxter equation and the R-matrix are central objects in the framework of quantum integrable systems
A different approach on the resolution of the spectrum of 1D statistical models is the Quantum Inverse Scattering (QISM) method, an elegant algebraic technique [50], that led directly to the invention of quasitriangular Hopf algebras known as quantum groups, which formally developed by Jimbo and Drinfeld independently [29,30,44,45]
Rump who developed a structure called a brace to describe all finite involutive set-theoretic solutions of the Yang–Baxter equation [63,64]. He showed that every brace provides a solution to the Yang–Baxter equation, and every non-degenerate, involutive set-theoretic solution of the Yang–Baxter equation can be obtained from a brace, a structure that generalizes nilpotent rings
Summary
The Yang–Baxter equation and the R-matrix are central objects in the framework of quantum integrable systems. We consider set-theoretic solutions of the Yang–Baxter and reflection equations coming from braces and we construct quantum spin chains with open boundary conditions through Sklyanin’s double row transfer matrix [65]. 5.2, more symmetries of open transfer matrices associated with certain classes of set-theoretic solutions of the Yang–Baxter equation coming from braces are discussed. The derivation of these symmetries is primarily based on the properties of the brace structures. Some of these symmetries generalize recent findings on periodic transfer matrices [26], while others are new. (3) In Sect. 5.3, symmetries of the double row transfer matrix constructed from the special class of Lyubashenko’s solutions are identified confirming some of the findings of Sect. 3
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