{"title":"Inductive description of quadratic Hom-Lie algebras with twist maps in the centroid","authors":"R. García-Delgado","doi":"arxiv-2409.04546","DOIUrl":null,"url":null,"abstract":"In this work we give an inductive way to construct quadratic Hom-Lie algebras\nwith twist maps in the centroid. We focus on those Hom-Lie algebras that are\nnot Lie algebras. We prove that the twist map of a Hom-Lie algebra of this type\nmust be nilpotent and the Hom-Lie algebra has trivial center. We also prove\nthat there exists a maximal ideal containing the kernel and the image of the\ntwist map. Then we state an inductive way to construct this type of Hom-Lie\nalgebras -- similar to the double extension procedure for Lie algebras -- and\nprove that any indecomposable quadratic Hom-Lie algebra with nilpotent twist\nmap in the centroid, which is not a Lie algebra, can be constructed using this\ntype of double extension.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"57 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-06","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-2409.04546","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this work we give an inductive way to construct quadratic Hom-Lie algebras
with twist maps in the centroid. We focus on those Hom-Lie algebras that are
not Lie algebras. We prove that the twist map of a Hom-Lie algebra of this type
must be nilpotent and the Hom-Lie algebra has trivial center. We also prove
that there exists a maximal ideal containing the kernel and the image of the
twist map. Then we state an inductive way to construct this type of Hom-Lie
algebras -- similar to the double extension procedure for Lie algebras -- and
prove that any indecomposable quadratic Hom-Lie algebra with nilpotent twist
map in the centroid, which is not a Lie algebra, can be constructed using this
type of double extension.