Invariant connections on non-irreducible symmetric spaces with simple Lie group

IF 0.8 3区数学 Q2 MATHEMATICS

Proceedings of the American Mathematical Society Pub Date : 2024-05-01 DOI:10.1090/proc/16903

Othmane Dani, Abdelhak Abouqateb, Saïd Benayadi

{"title":"Invariant connections on non-irreducible symmetric spaces with simple Lie group","authors":"Othmane Dani, Abdelhak Abouqateb, Saïd Benayadi","doi":"10.1090/proc/16903","DOIUrl":null,"url":null,"abstract":"<p>Consider a symmetric space <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G slash upper H\"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">G/H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with simple Lie group <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\"application/x-tex\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We demonstrate that when <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G slash upper H\"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">G/H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not irreducible, it is necessarily even dimensional and noncompact. Furthermore, the subgroup <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=\"application/x-tex\">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is also both noncompact and non-semisimple. Additionally, we establish that the only <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\"application/x-tex\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariant connection on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G slash upper H\"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">G/H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the canonical connection. On the other hand, we show that if <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G slash upper H\"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">G/H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has an odd dimension, it must be irreducible, and the subgroup <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=\"application/x-tex\">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> must be semisimple. Finally, we present an explicit example, and we show that there exists no other torsion-free <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\"application/x-tex\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariant connection on a symmetric space <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G slash upper H\"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">G/H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with semisimple Lie group <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\"application/x-tex\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which has the same curvature as the canonical one.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16903","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Consider a symmetric space G / H G/H with simple Lie group G G . We demonstrate that when G / H G/H is not irreducible, it is necessarily even dimensional and noncompact. Furthermore, the subgroup H H is also both noncompact and non-semisimple. Additionally, we establish that the only G G -invariant connection on G / H G/H is the canonical connection. On the other hand, we show that if G / H G/H has an odd dimension, it must be irreducible, and the subgroup H H must be semisimple. Finally, we present an explicit example, and we show that there exists no other torsion-free G G -invariant connection on a symmetric space G / H G/H with semisimple Lie group G G which has the same curvature as the canonical one.

查看原文本刊更多论文

具有简单李群的非还原对称空间上的不变连接

考虑具有简单李群 G G 的对称空间 G / H G/H 。我们证明，当 G / H G/H 不是不可还原时，它必然是偶数维和非紧密的。此外，子群 H H 也是非紧凑和非半复性的。此外，我们还确定了 G / H G/H 上唯一的 G G 不变连接是典型连接。另一方面，我们证明了如果 G / H G/H 的维数是奇数，那么它一定是不可还原的，子群 H H 一定是半简单的。最后，我们给出了一个明确的例子，并证明在具有半简单李群 G G 的对称空间 G / H G/H 上不存在其他与典型连接具有相同曲率的无扭 G G -不变连接。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Proceedings of the American Mathematical Society 数学-数学

CiteScore

1.70

自引率

10.00%

发文量

207

审稿时长

2-4 weeks

期刊介绍： All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to shorter research articles (not to exceed 15 printed pages) in all areas of pure and applied mathematics. To be published in the Proceedings, a paper must be correct, new, and significant. Further, it must be well written and of interest to a substantial number of mathematicians. Piecemeal results, such as an inconclusive step toward an unproved major theorem or a minor variation on a known result, are in general not acceptable for publication. Longer papers may be submitted to the Transactions of the American Mathematical Society. Published pages are the same size as those generated in the style files provided for AMS-LaTeX.