{"title":"Cup product, Frölicher-Nijenhuis bracket and the derived bracket associated to Hom-Lie algebras","authors":"Anusuiya Baishya, Apurba Das","doi":"arxiv-2409.01865","DOIUrl":null,"url":null,"abstract":"In this paper, we introduce some new graded Lie algebras associated with a\nHom-Lie algebra. At first, we define the cup product bracket and its\napplication to the deformation theory of Hom-Lie algebra morphisms. We observe\nan action of the well-known Hom-analogue of the Nijenhuis-Richardson graded Lie\nalgebra on the cup product graded Lie algebra. Using the corresponding\nsemidirect product, we define the Fr\\\"{o}licher-Nijenhuis bracket and study its\napplication to Nijenhuis operators. We show that the Nijenhuis-Richardson\ngraded Lie algebra and the Fr\\\"{o}licher-Nijenhuis algebra constitute a matched\npair of graded Lie algebras. Finally, we define another graded Lie bracket,\ncalled the derived bracket that is useful to study Rota-Baxter operators on\nHom-Lie algebras.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"51 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-03","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-2409.01865","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 paper, we introduce some new graded Lie algebras associated with a
Hom-Lie algebra. At first, we define the cup product bracket and its
application to the deformation theory of Hom-Lie algebra morphisms. We observe
an action of the well-known Hom-analogue of the Nijenhuis-Richardson graded Lie
algebra on the cup product graded Lie algebra. Using the corresponding
semidirect product, we define the Fr\"{o}licher-Nijenhuis bracket and study its
application to Nijenhuis operators. We show that the Nijenhuis-Richardson
graded Lie algebra and the Fr\"{o}licher-Nijenhuis algebra constitute a matched
pair of graded Lie algebras. Finally, we define another graded Lie bracket,
called the derived bracket that is useful to study Rota-Baxter operators on
Hom-Lie algebras.