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arXiv · 2003.08317

Set theoretic Yang-Baxter & reflection equations and quantum group symmetries

Anastasia Doikou·Agata Smoktunowicz
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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 ${\cal H}_N(q=1)$. 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 ${\cal B}_N(q=1, Q)$. 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.

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BibTeXRIS

Anastasia Doikou, Agata Smoktunowicz. 2020-03-18. Set theoretic Yang-Baxter & reflection equations and quantum group symmetries. https://doi.org/10.1007/s11005-021-01437-7

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