{"title":"复辛李群的强实伴随轨道","authors":"Tejbir Lohan , Chandan Maity","doi":"10.1016/j.laa.2024.11.015","DOIUrl":null,"url":null,"abstract":"<div><div>We consider the adjoint action of the symplectic Lie group <span><math><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> on its Lie algebra <span><math><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>. An element <span><math><mi>X</mi><mo>∈</mo><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> is called <span><math><msub><mrow><mi>Ad</mi></mrow><mrow><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></mrow></msub></math></span>-real if <span><math><mo>−</mo><mi>X</mi><mo>=</mo><mrow><mi>Ad</mi></mrow><mo>(</mo><mi>g</mi><mo>)</mo><mi>X</mi></math></span> for some <span><math><mi>g</mi><mo>∈</mo><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>. Moreover, if <span><math><mo>−</mo><mi>X</mi><mo>=</mo><mrow><mi>Ad</mi></mrow><mo>(</mo><mi>h</mi><mo>)</mo><mi>X</mi></math></span> for some involution <span><math><mi>h</mi><mo>∈</mo><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>, then <span><math><mi>X</mi><mo>∈</mo><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> is called strongly <span><math><msub><mrow><mi>Ad</mi></mrow><mrow><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></mrow></msub></math></span>-real. In this paper, we prove that for every element <span><math><mi>X</mi><mo>∈</mo><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>, there exists a skew-involution <span><math><mi>g</mi><mo>∈</mo><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> such that <span><math><mo>−</mo><mi>X</mi><mo>=</mo><mrow><mi>Ad</mi></mrow><mo>(</mo><mi>g</mi><mo>)</mo><mi>X</mi></math></span>. Furthermore, we classify the strongly <span><math><msub><mrow><mi>Ad</mi></mrow><mrow><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></mrow></msub></math></span>-real elements in <span><math><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>. We also classify skew-Hamiltonian matrices that are similar to their negatives via a symplectic involution.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"706 ","pages":"Pages 144-156"},"PeriodicalIF":1.0000,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Strongly real adjoint orbits of complex symplectic Lie group\",\"authors\":\"Tejbir Lohan , Chandan Maity\",\"doi\":\"10.1016/j.laa.2024.11.015\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We consider the adjoint action of the symplectic Lie group <span><math><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> on its Lie algebra <span><math><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>. An element <span><math><mi>X</mi><mo>∈</mo><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> is called <span><math><msub><mrow><mi>Ad</mi></mrow><mrow><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></mrow></msub></math></span>-real if <span><math><mo>−</mo><mi>X</mi><mo>=</mo><mrow><mi>Ad</mi></mrow><mo>(</mo><mi>g</mi><mo>)</mo><mi>X</mi></math></span> for some <span><math><mi>g</mi><mo>∈</mo><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>. Moreover, if <span><math><mo>−</mo><mi>X</mi><mo>=</mo><mrow><mi>Ad</mi></mrow><mo>(</mo><mi>h</mi><mo>)</mo><mi>X</mi></math></span> for some involution <span><math><mi>h</mi><mo>∈</mo><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>, then <span><math><mi>X</mi><mo>∈</mo><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> is called strongly <span><math><msub><mrow><mi>Ad</mi></mrow><mrow><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></mrow></msub></math></span>-real. In this paper, we prove that for every element <span><math><mi>X</mi><mo>∈</mo><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>, there exists a skew-involution <span><math><mi>g</mi><mo>∈</mo><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> such that <span><math><mo>−</mo><mi>X</mi><mo>=</mo><mrow><mi>Ad</mi></mrow><mo>(</mo><mi>g</mi><mo>)</mo><mi>X</mi></math></span>. Furthermore, we classify the strongly <span><math><msub><mrow><mi>Ad</mi></mrow><mrow><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></mrow></msub></math></span>-real elements in <span><math><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>. We also classify skew-Hamiltonian matrices that are similar to their negatives via a symplectic involution.</div></div>\",\"PeriodicalId\":18043,\"journal\":{\"name\":\"Linear Algebra and its Applications\",\"volume\":\"706 \",\"pages\":\"Pages 144-156\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-11-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Linear Algebra and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0024379524004348\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379524004348","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Strongly real adjoint orbits of complex symplectic Lie group
We consider the adjoint action of the symplectic Lie group on its Lie algebra . An element is called -real if for some . Moreover, if for some involution , then is called strongly -real. In this paper, we prove that for every element , there exists a skew-involution such that . Furthermore, we classify the strongly -real elements in . We also classify skew-Hamiltonian matrices that are similar to their negatives via a symplectic involution.
期刊介绍：
Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
