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Cartan decomposition

In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition or singular value decomposition of matrices. Its history can be traced to the 1880s work of Élie Cartan and Wilhelm Killing.

Art, Cartan involutions on Lie algebras & Cartan decomposition on the Lie group level

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Cartan involutions on Lie algebras

Killing formInvolutionAutomorphismPositive definite bilinear formInner automorphism
10 related topics

Cartan decomposition on the Lie group level

Analytic subgroupDiffeomorphismGeneral linear groupTotally geodesic
4 related topics

Cartan pairs

Distinct notion
1 related topics

Relation to polar decomposition

Invertible matrix
1 related topics

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Cartan involutions on Lie algebras

Cartan pairs

Cartan decomposition on the Lie group level

Relation to polar decomposition

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Cartan decomposition

Nodes30
Edges29
Triples10
Avg. degree1.93
Density0.066667
Components1

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Cartan decomposition

Top relations

related to Relation to polar decomposition · 8
Cartan decomposition → Cartan, Consider, Lie, SO, The, Then, Thus, Up
is a · 2
Cartan decomposition → decomposition of a semisimple Lie group or Lie algebra, polar decomposition of a matrix

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Important terminology

displaystyle mathfrak involution cartan lie theta algebra decomposition group semisimple identity involutions automorphism compact killing also polar diffeomorphism form map

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Cartan decompositionis adecomposition of a semisimple Lie group or Lie algebra0.90text
Cartan decompositionis apolar decomposition of a matrix0.90text
Cartan decompositionrelated to Relation to polar decompositionConsider0.60section
Cartan decompositionrelated to Relation to polar decompositionCartan0.60section
Cartan decompositionrelated to Relation to polar decompositionThen0.60section
Cartan decompositionrelated to Relation to polar decompositionLie0.60section
Cartan decompositionrelated to Relation to polar decompositionSO0.60section
Cartan decompositionrelated to Relation to polar decompositionThus0.60section
Cartan decompositionrelated to Relation to polar decompositionUp0.60section
Cartan decompositionrelated to Relation to polar decompositionThe0.60section

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