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The Cartan and Iwasawa decompositions

Juan Manuel Lorenzo Naveiro

Universidade de Santiago de Compostela

Seminar on Symmetric Spaces, 2022

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Symmetric spaces

  •  

 

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Symmetric spaces

  •  

 

Symmetric spaces are

complete.

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Symmetric spaces

  •  

Symmetric spaces are

complete.

 

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Symmetric spaces

  •  

Symmetric spaces are

homogeneous.

 

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Symmetric spaces

  •  

Symmetric spaces are

homogeneous.

 

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Symmetric spaces

  •  

Symmetric spaces are

homogeneous.

 

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From geometry to algebra

 

  •  

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From geometry to algebra

 

  •  

Definition

 

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From algebra to geometry

 

 

 

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From algebra to geometry

 

 

 

 

 

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From algebra to geometry

 

 

 

 

 

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The Cartan decomposition

  •  

Definition

 

 

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Connection and curvature

Proposition

 

Proof

 

 

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Connection and curvature

Proposition

 

Proof

 

 

 

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Connection and curvature

Proposition

 

Proof

 

 

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Connection and curvature

Proposition

 

Proof

 

 

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Connection and curvature

Proposition

 

Proof

 

 

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Connection and curvature

 

Proposition

 

Theorem

 

 

 

 

 

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Type

  •  

 

 

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Type

  •  

 

Compact type

    • negative definite.
    • compact, semisimple.
    • .

Noncompact type

    • negative definite.
    • noncompact, semisimple.
    • .

 

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Noncompact symmetric spaces

 

 

 

 

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Noncompact symmetric spaces

 

Theorem

M is a Hadamard manifold.

Proof

 

 

 

 

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Noncompact symmetric spaces

 

Theorem

M is a Hadamard manifold.

Proof

 

 

 

 

 

 

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Noncompact symmetric spaces

 

Theorem

 

Proof

 

 

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Noncompact symmetric spaces

 

Theorem

 

Proof

 

 

 

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Root space decomposition

  •  

 

(Root space)

 

(Roots)

 

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Properties of root spaces

 

 

 

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Properties of root spaces

 

 

 

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Properties of root spaces

 

 

 

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Properties of root spaces

 

 

 

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Properties of root spaces

 

 

 

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Properties of root spaces

 

 

 

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Properties of root spaces

 

 

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Properties of root spaces

 

 

 

 

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Properties of root spaces

 

 

 

 

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Positive roots

  •  

 

Definition

 

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Iwasawa decomposition

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Theorem (Iwasawa decomposition, Lie algebra level)

 

 

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Theorem (Iwasawa decomposition, Lie algebra level)

 

Key example

 

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Theorem (Iwasawa decomposition, Lie algebra level)

 

Proof

 

 

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Theorem (Iwasawa decomposition, Lie algebra level)

 

Proof

 

 

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Theorem (Iwasawa decomposition, Lie algebra level)

 

Proof

 

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Theorem (Iwasawa decomposition, Lie algebra level)

 

Proof

 

 

 

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Theorem (Iwasawa decomposition, Lie algebra level)

 

Proof

 

 

 

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Theorem (Iwasawa decomposition, Lie algebra level)

 

Proof

 

 

 

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Iwasawa decomposition

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Theorem (Iwasawa decomposition, Lie group level)

 

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Theorem (Iwasawa decomposition, Lie group level)

 

Proof (sketch)

Under a suitable basis:

 

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Proof (sketch)

 

Theorem (Iwasawa decomposition, Lie group level)

 

 

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Theorem (Iwasawa decomposition, Lie group level)

 

Proof (sketch)

 

 

 

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The solvable model

  •  

 

is a diffeomorphism. The pullback metric is left-invariant.

Theorem

 

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The solvable model

  •  

 

is a diffeomorphism. The pullback metric is left-invariant.

 

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