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.

arXiv comment38 pages, Latex. Various clarifying comments introduced, two references added. Version to appear in LMP

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Cited by in corpus (18)

• Introduction to classical and quantum integrability
• Set theoretic Yang-Baxter equation, braces and Drinfeld twists
• From Braces to Hecke algebras & Quantum Groups
• Soliton interactions and Yang-Baxter maps for the complex coupled short-pulse equation
• Quandles as pre-Lie skew braces, set-theoretic Hopf algebras & universal R-matrices
• Near braces and p-deformed braided groups
• Two-component Yang-Baxter maps and star-triangle relations
• Reflection equation as a tool for studying solutions to the Yang-Baxter equation
• Skew braces: a brief survey
• Discretizations of the generalized AKNS scheme
• Novel non-involutive solutions of the Yang-Baxter equation from (skew) braces
• The Yang-Baxter equation, Quantum computing and Quantum entanglement
• Reflections to set-theoretic solutions of the Yang-Baxter equation
• Superintegrable cellular automata and dual unitary gates from Yang-Baxter maps
• Reflection Maps Associated with Involutions and Factorization Problems, and Their Poisson Geometry
• Algebraic structures in set-theoretic Yang-Baxter & reflection equations
• Self-distributive structures, braces & the Yang-Baxter equation
• Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation