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In and around the origin of quantum groups. (English) Zbl 1129.81046

Mourão, José C. (ed.) et al., Prospects in mathematical physics. Young researchers symposium of the 14th international congress on mathematical physics, Lisbon, Portugal, July 25–26, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4270-6/pbk). Contemporary Mathematics 435, 101-126 (2007).
Summary: 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 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.
For the entire collection see [Zbl 1119.81006].

MSC:

81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R60 Noncommutative geometry in quantum theory
46L60 Applications of selfadjoint operator algebras to physics
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