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R-Matrices, Yetter-Drinfel'd Modules and Yang-Baxter Equation

Institut de Mathématiques de Jussieu–Paris Rive Gauche, UMR7586, Bâtiment Sophie Germain, Case 7012, 75205 PARIS Cedex 13, France
Axioms 2013, 2(3), 443-476;
Received: 14 August 2013 / Revised: 28 August 2013 / Accepted: 30 August 2013 / Published: 5 September 2013
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2013)
PDF [402 KB, uploaded 5 September 2013]


In the first part we recall two famous sources of solutions to the Yang-Baxter equation—R-matrices and Yetter-Drinfel0d (=YD) modules—and an interpretation of the former as a particular case of the latter. We show that this result holds true in the more general case of weak R-matrices, introduced here. In the second part we continue exploring the “braided” aspects of YD module structure, exhibiting a braided system encoding all the axioms from the definition of YD modules. The functoriality and several generalizations of this construction are studied using the original machinery of YD systems. As consequences, we get a conceptual interpretation of the tensor product structures for YD modules, and a generalization of the deformation cohomology of YD modules. This homology theory is thus included into the unifying framework of braided homologies, which contains among others Hochschild, Chevalley-Eilenberg, Gerstenhaber-Schack and quandle homologies. View Full-Text
Keywords: Yang-Baxter equation; braided system; Yetter-Drinfel'd module; R-matrix; braided homology Yang-Baxter equation; braided system; Yetter-Drinfel'd module; R-matrix; braided homology

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Lebed, V. R-Matrices, Yetter-Drinfel'd Modules and Yang-Baxter Equation. Axioms 2013, 2, 443-476.

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