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Quasitriangular Hopf algebras and Yang-Baxter equations. (English) Zbl 0709.17009

Summary: This expository article is intended as an informal introduction to the theory of quasitriangular Hopf algebras and its connections with physics. Basic properties and applications of Hopf algebras and Yang-Baxter equations are reviewed, with the quantum group \(U_ q(sl_ 2)\) as a frequent example. The development builds up to the representation theory of quasitriangular Hopf algebras. Much of the abstract representation theory is new, including a formula for the rank of a representation.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
16G99 Representation theory of associative rings and algebras
22E70 Applications of Lie groups to the sciences; explicit representations
57T05 Hopf algebras (aspects of homology and homotopy of topological groups)
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
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