The Yang–baxter Equation: Mathematical Structures, Physical Realizations, and Applications

Author(s)	Dr. Rakesh Paswan, Ms. Deep Prabha Bharti
Country	India
Abstract	The Yang–Baxter Equation (YBE) is a central algebraic identity that governs the compatibility of multi-body interactions in both mathematics and physics. Originating in the context of factorized scattering and exactly solvable models, the YBE has evolved into a unifying principle connecting integrable systems, representation theory, low-dimensional topology, braid groups, quantum groups, tensor categories, and topological quantum computation. This article provides a comprehensive exposition of the YBE, bridging its mathematical foundations with its physical applications. We develop constant and spectral-parameter forms, construct explicit solutions, introduce diagrammatic and tensor-network interpretations, and explore the role of the YBE in integrable models, quantum symmetries, and topological phases of matter.
Keywords	Yang–Baxter Equation, integrable systems, R-matrix, quantum groups, braid groups, solvable models, topological quantum computation
Field	Mathematics > Maths + Physics
Published In	Volume 17, Issue 1, January-March 2026
Published On	2026-02-18

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