{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:32:33Z","timestamp":1760243553674,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2013,9,5]],"date-time":"2013-09-05T00:00:00Z","timestamp":1378339200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In the first part we recall two famous sources of solutions to the Yang-Baxter equation\u2014R-matrices and Yetter-Drinfel0d (=YD) modules\u2014and 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 \u201cbraided\u201d 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.<\/jats:p>","DOI":"10.3390\/axioms2030443","type":"journal-article","created":{"date-parts":[[2013,9,5]],"date-time":"2013-09-05T10:54:54Z","timestamp":1378378494000},"page":"443-476","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["R-Matrices, Yetter-Drinfel'd Modules and Yang-Baxter Equation"],"prefix":"10.3390","volume":"2","author":[{"given":"Victoria","family":"Lebed","sequence":"first","affiliation":[{"name":"Institut de Math\u00e9matiques de Jussieu\u2013Paris Rive Gauche, UMR7586, B\u00e2timent Sophie Germain, Case 7012, 75205 PARIS Cedex 13, France"}]}],"member":"1968","published-online":{"date-parts":[[2013,9,5]]},"reference":[{"key":"ref_1","unstructured":"Drinfel\u2019d, V.G. (,  1986). Quantum Groups. 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