{"title":"近关联代数和 $LR$ 代数的代数理论:等价、表征和 $LR$ 扬-巴克斯特方程","authors":"Elisabete Barreiro, Saïd Benayadi, Carla Rizzo","doi":"arxiv-2409.00390","DOIUrl":null,"url":null,"abstract":"We develop the bialgebra theory for two classes of non-associative algebras:\nnearly associative algebras and $LR$-algebras. In particular, building on\nrecent studies that reveal connections between these algebraic structures, we\nestablish that nearly associative bialgebras and $LR$-bialgebras are, in fact,\nequivalent concepts. We also provide a characterization of these bialgebra\nclasses based on the coproduct. Moreover, since the development of nearly\nassociative bialgebras - and by extension, $LR$-bialgebras - requires the\nframework of nearly associative $L$-algebras, we introduce this class of\nnon-associative algebras and explore their fundamental properties. Furthermore,\nwe identify and characterize a special class of nearly associative bialgebras,\nthe coboundary nearly associative bialgebras, which provides a natural\nframework for studying the Yang-Baxter equation (YBE) within this context.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"14 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Bialgebra theory for nearly associative algebras and $LR$-algebras: equivalence, characterization, and $LR$-Yang-Baxter Equation\",\"authors\":\"Elisabete Barreiro, Saïd Benayadi, Carla Rizzo\",\"doi\":\"arxiv-2409.00390\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We develop the bialgebra theory for two classes of non-associative algebras:\\nnearly associative algebras and $LR$-algebras. In particular, building on\\nrecent studies that reveal connections between these algebraic structures, we\\nestablish that nearly associative bialgebras and $LR$-bialgebras are, in fact,\\nequivalent concepts. We also provide a characterization of these bialgebra\\nclasses based on the coproduct. Moreover, since the development of nearly\\nassociative bialgebras - and by extension, $LR$-bialgebras - requires the\\nframework of nearly associative $L$-algebras, we introduce this class of\\nnon-associative algebras and explore their fundamental properties. Furthermore,\\nwe identify and characterize a special class of nearly associative bialgebras,\\nthe coboundary nearly associative bialgebras, which provides a natural\\nframework for studying the Yang-Baxter equation (YBE) within this context.\",\"PeriodicalId\":501136,\"journal\":{\"name\":\"arXiv - MATH - Rings and Algebras\",\"volume\":\"14 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Rings and Algebras\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.00390\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.00390","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Bialgebra theory for nearly associative algebras and $LR$-algebras: equivalence, characterization, and $LR$-Yang-Baxter Equation
We develop the bialgebra theory for two classes of non-associative algebras:
nearly associative algebras and $LR$-algebras. In particular, building on
recent studies that reveal connections between these algebraic structures, we
establish that nearly associative bialgebras and $LR$-bialgebras are, in fact,
equivalent concepts. We also provide a characterization of these bialgebra
classes based on the coproduct. Moreover, since the development of nearly
associative bialgebras - and by extension, $LR$-bialgebras - requires the
framework of nearly associative $L$-algebras, we introduce this class of
non-associative algebras and explore their fundamental properties. Furthermore,
we identify and characterize a special class of nearly associative bialgebras,
the coboundary nearly associative bialgebras, which provides a natural
framework for studying the Yang-Baxter equation (YBE) within this context.