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

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.

38 pages, Latex. Various clarifying comments introduced, two references added. Version to appear in LMP

References in corpus (4)

• Set-theoretical reflection equation: Classification of reflection maps
• Set theoretic Yang-Baxter equation, braces and Drinfeld twists
• $A_n^{(1)}$ affine Toda field theories with integrable boundary conditions revisited
• (Quantum) twisted Yangians: symmetry, Baxterisation and centralizers

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