Left-Invariant Einstein-like Metrics on Compact Lie Groups
In this paper, we study left-invariant Einstein-like metrics on the compact Lie group <i>G</i>. Assume that there exist two subgroups, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>...
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MDPI AG
2022-05-01
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| Online Access: | https://www.mdpi.com/2227-7390/10/9/1510 |
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| author | An Wu Huafei Sun |
| author_facet | An Wu Huafei Sun |
| author_sort | An Wu |
| collection | DOAJ |
| description | In this paper, we study left-invariant Einstein-like metrics on the compact Lie group <i>G</i>. Assume that there exist two subgroups, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>⊂</mo><mi>K</mi><mo>⊂</mo><mi>G</mi></mrow></semantics></math></inline-formula>, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>/</mo><mi>K</mi></mrow></semantics></math></inline-formula> is a compact, connected, irreducible, symmetric space, and the isotropy representation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>/</mo><mi>H</mi></mrow></semantics></math></inline-formula> has exactly two inequivalent, irreducible summands. We prove that the left metric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>⟨</mo><mo>·</mo><mo>,</mo><mo>·</mo><mo>⟩</mo></mrow><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub></mrow></msub></semantics></math></inline-formula> on <i>G</i> defined by the first equation, must be an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>-metric. Moreover, we prove that compact Lie groups do not admit non-naturally reductive left-invariant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">B</mi></semantics></math></inline-formula>-metrics, such as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>⟨</mo><mo>·</mo><mo>,</mo><mo>·</mo><mo>⟩</mo></mrow><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub></mrow></msub></semantics></math></inline-formula>. |
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| spelling | doaj.art-1f4acdc6a82c4e38a3ca8f9d6297589f2023-11-23T08:45:26ZengMDPI AGMathematics2227-73902022-05-01109151010.3390/math10091510Left-Invariant Einstein-like Metrics on Compact Lie GroupsAn Wu0Huafei Sun1School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 102488, ChinaSchool of Mathematics and Statistics, Beijing Institute of Technology, Beijing 102488, ChinaIn this paper, we study left-invariant Einstein-like metrics on the compact Lie group <i>G</i>. Assume that there exist two subgroups, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>⊂</mo><mi>K</mi><mo>⊂</mo><mi>G</mi></mrow></semantics></math></inline-formula>, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>/</mo><mi>K</mi></mrow></semantics></math></inline-formula> is a compact, connected, irreducible, symmetric space, and the isotropy representation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>/</mo><mi>H</mi></mrow></semantics></math></inline-formula> has exactly two inequivalent, irreducible summands. We prove that the left metric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>⟨</mo><mo>·</mo><mo>,</mo><mo>·</mo><mo>⟩</mo></mrow><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub></mrow></msub></semantics></math></inline-formula> on <i>G</i> defined by the first equation, must be an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>-metric. Moreover, we prove that compact Lie groups do not admit non-naturally reductive left-invariant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">B</mi></semantics></math></inline-formula>-metrics, such as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>⟨</mo><mo>·</mo><mo>,</mo><mo>·</mo><mo>⟩</mo></mrow><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub></mrow></msub></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/10/9/1510homogeneous spacecompact Lie groupsEinstein-like metric\({\mathcal{A}}\)-metricℬ-metric |
| spellingShingle | An Wu Huafei Sun Left-Invariant Einstein-like Metrics on Compact Lie Groups Mathematics homogeneous space compact Lie groups Einstein-like metric \({\mathcal{A}}\)-metric ℬ-metric |
| title | Left-Invariant Einstein-like Metrics on Compact Lie Groups |
| title_full | Left-Invariant Einstein-like Metrics on Compact Lie Groups |
| title_fullStr | Left-Invariant Einstein-like Metrics on Compact Lie Groups |
| title_full_unstemmed | Left-Invariant Einstein-like Metrics on Compact Lie Groups |
| title_short | Left-Invariant Einstein-like Metrics on Compact Lie Groups |
| title_sort | left invariant einstein like metrics on compact lie groups |
| topic | homogeneous space compact Lie groups Einstein-like metric \({\mathcal{A}}\)-metric ℬ-metric |
| url | https://www.mdpi.com/2227-7390/10/9/1510 |
| work_keys_str_mv | AT anwu leftinvarianteinsteinlikemetricsoncompactliegroups AT huafeisun leftinvarianteinsteinlikemetricsoncompactliegroups |