{"title":"Nijenhuis operators with a unity and \n \n F\n $F$\n -manifolds","authors":"Evgenii I. Antonov, Andrey Yu. Konyaev","doi":"10.1112/jlms.12983","DOIUrl":null,"url":null,"abstract":"<p>The core object of this paper is a pair <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>L</mi>\n <mo>,</mo>\n <mi>e</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$(L, e)$</annotation>\n </semantics></math>, where <span></span><math>\n <semantics>\n <mi>L</mi>\n <annotation>$L$</annotation>\n </semantics></math> is a Nijenhuis operator and <span></span><math>\n <semantics>\n <mi>e</mi>\n <annotation>$e$</annotation>\n </semantics></math> is a vector field satisfying a specific Lie derivative condition, that is, <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>L</mi>\n <mi>e</mi>\n </msub>\n <mi>L</mi>\n <mo>=</mo>\n <mo>Id</mo>\n </mrow>\n <annotation>$\\mathcal {L}_{e}L=\\operatorname{Id}$</annotation>\n </semantics></math>. Our research unfolds in two parts. In the first part, we establish a splitting theorem for Nijenhuis operators with a unity, offering an effective reduction of their study to cases where <span></span><math>\n <semantics>\n <mi>L</mi>\n <annotation>$L$</annotation>\n </semantics></math> has either one real or two complex conjugate eigenvalues at a given point. We further provide the normal forms for <span></span><math>\n <semantics>\n <mi>gl</mi>\n <annotation>$\\mathrm{gl}$</annotation>\n </semantics></math>-regular Nijenhuis operators with a unity around algebraically generic points, along with seminormal forms for dimensions 2 and 3. In the second part, we establish the relationship between Nijenhuis operators with a unity and <span></span><math>\n <semantics>\n <mi>F</mi>\n <annotation>$F$</annotation>\n </semantics></math>-manifolds. Specifically, we prove that the class of regular <span></span><math>\n <semantics>\n <mi>F</mi>\n <annotation>$F$</annotation>\n </semantics></math>-manifolds coincides with the class of Nijenhuis manifolds with a cyclic unity. Extending our results from dimension 3, we reveal seminormal forms for corresponding <span></span><math>\n <semantics>\n <mi>F</mi>\n <annotation>$F$</annotation>\n </semantics></math>-manifolds around singularities.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12983","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12983","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The core object of this paper is a pair , where is a Nijenhuis operator and is a vector field satisfying a specific Lie derivative condition, that is, . Our research unfolds in two parts. In the first part, we establish a splitting theorem for Nijenhuis operators with a unity, offering an effective reduction of their study to cases where has either one real or two complex conjugate eigenvalues at a given point. We further provide the normal forms for -regular Nijenhuis operators with a unity around algebraically generic points, along with seminormal forms for dimensions 2 and 3. In the second part, we establish the relationship between Nijenhuis operators with a unity and -manifolds. Specifically, we prove that the class of regular -manifolds coincides with the class of Nijenhuis manifolds with a cyclic unity. Extending our results from dimension 3, we reveal seminormal forms for corresponding -manifolds around singularities.
本文的核心对象是一对 ( L , e ) $(L, e)$ ,其中 L $L$ 是一个尼延胡斯算子,e $e$ 是一个满足特定列导数条件的向量场,即 L e L = Id $\mathcal {L}_{e}L=\operatorname{Id}$ 。我们的研究分两部分展开。在第一部分中,我们建立了具有一元性的尼延胡斯算子的分裂定理,从而将其研究有效地简化为 L $L$ 在给定点上具有一个实共轭特征值或两个复共轭特征值的情况。我们还进一步提供了在代数通项点周围具有一元性的 Gl $\mathrm{gl}$ 不规则尼延胡斯算子的正则形式,以及维数 2 和维数 3 的半正则形式。在第二部分,我们建立了具有统一性的尼延胡伊斯算子与 F $F$ -manifolds 之间的关系。具体地说,我们证明了正则 F $F$ -manifolds 类与具有循环统一性的尼延胡斯流形类重合。通过扩展维 3 的结果,我们揭示了奇点周围相应 F $F$ -manifold 的半正态形式。
期刊介绍:
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.