{"title":"论后李代数的普遍包络代数的子邻接霍普夫代数","authors":"Yunnan Li","doi":"arxiv-2408.01345","DOIUrl":null,"url":null,"abstract":"Recently the notion of post-Hopf algebra was introduced, with the universal\nenveloping algebra of a post-Lie algebra as the fundamental example. A novel\nproperty is that any cocommutative post-Hopf algebra gives rise to a\nsub-adjacent Hopf algebra with a generalized Grossman-Larson product. By\ntwisting the post-Hopf product, we provide a combinatorial antipode formula for\nthe sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie\nalgebra. Relating to such a sub-adjacent Hopf algebra, we also obtain a closed\ninverse formula for the Oudom-Guin isomorphism in the context of post-Lie\nalgebras. Especially as a byproduct, we derive a cancellation-free antipode\nformula for the Grossman-Larson Hopf algebra of ordered trees through a\nconcrete tree-grafting expression.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"9 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra\",\"authors\":\"Yunnan Li\",\"doi\":\"arxiv-2408.01345\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Recently the notion of post-Hopf algebra was introduced, with the universal\\nenveloping algebra of a post-Lie algebra as the fundamental example. A novel\\nproperty is that any cocommutative post-Hopf algebra gives rise to a\\nsub-adjacent Hopf algebra with a generalized Grossman-Larson product. By\\ntwisting the post-Hopf product, we provide a combinatorial antipode formula for\\nthe sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie\\nalgebra. Relating to such a sub-adjacent Hopf algebra, we also obtain a closed\\ninverse formula for the Oudom-Guin isomorphism in the context of post-Lie\\nalgebras. Especially as a byproduct, we derive a cancellation-free antipode\\nformula for the Grossman-Larson Hopf algebra of ordered trees through a\\nconcrete tree-grafting expression.\",\"PeriodicalId\":501136,\"journal\":{\"name\":\"arXiv - MATH - Rings and Algebras\",\"volume\":\"9 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-02\",\"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-2408.01345\",\"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-2408.01345","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra
Recently the notion of post-Hopf algebra was introduced, with the universal
enveloping algebra of a post-Lie algebra as the fundamental example. A novel
property is that any cocommutative post-Hopf algebra gives rise to a
sub-adjacent Hopf algebra with a generalized Grossman-Larson product. By
twisting the post-Hopf product, we provide a combinatorial antipode formula for
the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie
algebra. Relating to such a sub-adjacent Hopf algebra, we also obtain a closed
inverse formula for the Oudom-Guin isomorphism in the context of post-Lie
algebras. Especially as a byproduct, we derive a cancellation-free antipode
formula for the Grossman-Larson Hopf algebra of ordered trees through a
concrete tree-grafting expression.