Set-theoretic Yang–Baxter & reflection equations and quantum group symmetries
Anastasia Doikou, Agata Smoktunowicz
Abstract
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 $${\mathcal {H}}_N(q=1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . 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 $${\mathcal {B}}_N(q=1, Q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . 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.