{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:52:06Z","timestamp":1760233926426,"version":"build-2065373602"},"reference-count":49,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,12]],"date-time":"2021-03-12T00:00:00Z","timestamp":1615507200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006445","name":"University of Warsaw","doi-asserted-by":"publisher","award":["It does not apply"],"award-info":[{"award-number":["It does not apply"]}],"id":[{"id":"10.13039\/501100006445","id-type":"DOI","asserted-by":"publisher"}]},{"name":"University of Warsaw and Jagiellonian University (Kartezjusz program)","award":["It does not apply"],"award-info":[{"award-number":["It does not apply"]}]},{"name":"Narodowe Centrum Nauki (POLAND)","award":["2016\/22\/M\/ST1\/00542"],"award-info":[{"award-number":["2016\/22\/M\/ST1\/00542"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This work introduces a new concept, the so-called Darboux family, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie algebra automorphisms, in a relatively easy manner. The Darboux family notion can be considered as a generalisation of the Darboux polynomial for a vector field. The classification of r-matrices and solutions to classical Yang\u2013Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail. Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix representations of certain Lie algebras with a non-trivial center is developed.<\/jats:p>","DOI":"10.3390\/sym13030465","type":"journal-article","created":{"date-parts":[[2021,3,15]],"date-time":"2021-03-15T02:51:48Z","timestamp":1615776708000},"page":"465","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8643-144X","authenticated-orcid":false,"given":"Javier","family":"de Lucas","sequence":"first","affiliation":[{"name":"Department of Mathematical Methods in Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daniel","family":"Wysocki","sequence":"additional","affiliation":[{"name":"Department of Mathematical Methods in Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,12]]},"reference":[{"key":"ref_1","unstructured":"Chari, V., and Pressley, P. (1994). A Guide to Quantum Groups, Cambridge University Press."},{"key":"ref_2","first-page":"68","article-title":"Hamiltonian structures of Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equation","volume":"27","author":"Drinfeld","year":"1983","journal-title":"Sov. Math. Dokl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"898","DOI":"10.1007\/BF01247086","article-title":"Quantum groups","volume":"41","author":"Drinfeld","year":"1986","journal-title":"J. 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