Left-Invariant Einstein-like Metrics on Compact Lie Groups

In this paper, we study left-invariant Einstein-like metrics on the compact Lie group G. Assume that there exist two subgroups, H⊂K⊂G, such that G/K is a compact, connected, irreducible, symmetric space, and the isotropy representation of G/H has exactly two inequivalent, irreducible summands. We pr...

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Published inMathematics (Basel) Vol. 10; no. 9; p. 1510
Main Authors Wu, An, Sun, Huafei
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.05.2022
Subjects
({\mathcal{A}}\)-metric
compact Lie groups
Decomposition
Einstein-like metric
homogeneous space
Invariants
Isotropy
Lie groups
Mathematics
Subgroups
ℬ-metric
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Summary:In this paper, we study left-invariant Einstein-like metrics on the compact Lie group G. Assume that there exist two subgroups, H⊂K⊂G, such that G/K is a compact, connected, irreducible, symmetric space, and the isotropy representation of G/H has exactly two inequivalent, irreducible summands. We prove that the left metric ⟨·,·⟩t1,t2 on G defined by the first equation, must be an A-metric. Moreover, we prove that compact Lie groups do not admit non-naturally reductive left-invariant B-metrics, such as ⟨·,·⟩t1,t2.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math10091510