{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:53:49Z","timestamp":1760147629297,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,2,21]],"date-time":"2023-02-21T00:00:00Z","timestamp":1676937600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Seminar Alexander Zavadskij on Representation of Algebras and their Applications, Universidad Nacional de Colombia"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>The Kronecker algebra K is the path algebra induced by the quiver with two parallel arrows, one source and one sink (i.e., a quiver with two vertices and two arrows going in the same direction). Modules over K are said to be Kronecker modules. The classification of these modules can be obtained by solving a well-known tame matrix problem. Such a classification deals with solving systems of differential equations of the form Ax=Bx\u2032, where A and B are m\u00d7n, F-matrices with F an algebraically closed field. On the other hand, researching the Yang\u2013Baxter equation (YBE) is a topic of great interest in several science fields. It has allowed advances in physics, knot theory, quantum computing, cryptography, quantum groups, non-associative algebras, Hopf algebras, etc. It is worth noting that giving a complete classification of the YBE solutions is still an open problem. This paper proves that some indecomposable modules over K called pre-injective Kronecker modules give rise to some algebraic structures called skew braces which allow the solutions of the YBE. Since preprojective Kronecker modules categorize some integer sequences via some appropriated snake graphs, we prove that such modules are automatic and that they induce the automatic sequences of continued fractions.<\/jats:p>","DOI":"10.3390\/computation11030043","type":"journal-article","created":{"date-parts":[[2023,2,22]],"date-time":"2023-02-22T03:32:17Z","timestamp":1677036737000},"page":"43","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Solutions of the Yang\u2013Baxter Equation and Automaticity Related to Kronecker Modules"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6812-5131","authenticated-orcid":false,"given":"Agust\u00edn Moreno","family":"Ca\u00f1adas","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad Nacional de Colombia, Edificio Yu Takeuchi 404, Kra 30 No 45-03, Bogot\u00e1 11001000, Colombia"}]},{"given":"Pedro Fernando","family":"Fern\u00e1ndez Espinosa","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad de Caldas, Cll 65 No 26-10, Manizales 170004, Colombia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2051-9075","authenticated-orcid":false,"given":"Adolfo","family":"Ballester-Bolinches","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat de Val\u00e8ncia, Dr. Moliner 50, Burjassot, 46100 Val\u00e8ncia, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1312","DOI":"10.1103\/PhysRevLett.19.1312","article-title":"Some exact results for the many-body problem in one dimension with repulsive delta-function interaction","volume":"19","author":"Yang","year":"1967","journal-title":"Phys. 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