{"title":"Invariant Metrics on Nilpotent Lie algebras","authors":"R. García-Delgado","doi":"arxiv-2409.09017","DOIUrl":null,"url":null,"abstract":"We state criteria for a nilpotent Lie algebra $\\g$ to admit an invariant\nmetric. We use that $\\g$ possesses two canonical abelian ideals $\\ide(\\g)\n\\subset \\mathfrak{J}(\\g)$ to decompose the underlying vector space of $\\g$ and\nthen we state sufficient conditions for $\\g$ to admit an invariant metric. The\nproperties of the ideal $\\mathfrak{J}(\\g)$ allows to prove that if a current\nLie algebra $\\g \\otimes \\Sa$ admits an invariant metric, then there must be an\ninvariant and non-degenerate bilinear map from $\\Sa \\times \\Sa$ into the space\nof centroids of $\\g/\\mathfrak{J}(\\g)$. We also prove that in any nilpotent Lie\nalgebra $\\g$ there exists a non-zero, symmetric and invariant bilinear form.\nThis bilinear form allows to reconstruct $\\g$ by means of an algebra with unit.\nWe prove that this algebra is simple if and only if the bilinear form is an\ninvariant metric on $\\g$.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"6 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-13","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-2409.09017","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
We state criteria for a nilpotent Lie algebra $\g$ to admit an invariant
metric. We use that $\g$ possesses two canonical abelian ideals $\ide(\g)
\subset \mathfrak{J}(\g)$ to decompose the underlying vector space of $\g$ and
then we state sufficient conditions for $\g$ to admit an invariant metric. The
properties of the ideal $\mathfrak{J}(\g)$ allows to prove that if a current
Lie algebra $\g \otimes \Sa$ admits an invariant metric, then there must be an
invariant and non-degenerate bilinear map from $\Sa \times \Sa$ into the space
of centroids of $\g/\mathfrak{J}(\g)$. We also prove that in any nilpotent Lie
algebra $\g$ there exists a non-zero, symmetric and invariant bilinear form.
This bilinear form allows to reconstruct $\g$ by means of an algebra with unit.
We prove that this algebra is simple if and only if the bilinear form is an
invariant metric on $\g$.