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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
Received: 14 August 2013; in revised form: 28 August 2013 / Accepted: 30 August 2013 / Published: 5 September 2013
Abstract: 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.
Keywords: Yang-Baxter equation; braided system; Yetter-Drinfel'd module; R-matrix; braided homology
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MDPI and ACS Style
Lebed, V. R-Matrices, Yetter-Drinfel'd Modules and Yang-Baxter Equation. Axioms 2013, 2, 443-476.
Lebed V. R-Matrices, Yetter-Drinfel'd Modules and Yang-Baxter Equation. Axioms. 2013; 2(3):443-476.
Lebed, Victoria. 2013. "R-Matrices, Yetter-Drinfel'd Modules and Yang-Baxter Equation." Axioms 2, no. 3: 443-476.