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arXiv:math/0309199

In and around the origin of quantum groups

Vaughan F. R. Jones

Published Sep 11, 2003• 0 citations
arXivHTMLPDF

Abstract

Quantum groups were invented largely to provide solutions of the Yang-Baxter equation and hence solvable models in 2-dimensional statistical mechanics and one-dimensional quantum mechanics. They have been hugely successful. But not all Yang-Baxter solutions fit into the framework of quantum groups. We shall explain how other mathematical structures, especially subfactors, provide a language and examples for solvable models. The prevalence of the Connes tensor product of Hilbert spaces over von Neumann algebras leads us to speculate concerning its potential role in describing entangled or interacting quantum systems.

DOI
10.48550/arxiv.math/0309199

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Quantum groups were invented largely to provide solutions of the Yang-Baxter equation and hence solvable models in 2-dimensional statistical mechanics and one-dimensional quantum mechanics. They have been hugely successful. But not all Yang-Baxter solutions fit into the framework of quantum groups. We shall explain how other mathematical structures, especially subfactors, provide a language and examples for solvable models. The prevalence of the Connes tensor product of Hilbert spaces over von Neumann algebras leads us to speculate concerning its potential role in describing entangled or interacting quantum systems.
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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.
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This review is an expansion of my talk at the conference on Sixty Years of Yang–Mills Theory. I review and explain the line of thoughts that lead to a recent joint work with Hu and Geer [Hu et al., arXiv:1502.03433] on the construction, exact solutions and ubiquitous properties of a class of quantum group gauge models on a honey-comb lattice. Conceptually the construction achieves a synthesis of the ideas of Yang–Baxter equations with those of Yang–Mills theory. Physically the models describe topological anyonic states in 2D systems.