Lie theory · Real form

$\mathfrak{e}_{6}\;\text{compact}$ e₆ compact real form

Compact real form of e₆(C) with its curated integrating groups.

Reference coverageCompact real form

Definition

The compact real form of the complex simple Lie algebra e₆(C).

Definition sources

Real dimension: $78$

Complexification

\mathfrak{e}_{6}(\mathbb C)

This is the compact real form. The integration links below identify particular global groups with this local algebra.

Reference properties

Property	Value	Evidence
Real dimension	$78$	Sources
Complexification	$\mathfrak{e}_{6}(\mathbb C)$	Sources
Compact real form	Yes	Sources
Real rank	$0$	Sources

One algebra, several integrations

A real form is algebraic data over R. The linked group records fix particular connected integrations, and state their global topology separately. Compactness of this real algebra should not be used to infer a particular quotient kernel. [sources]

Related objects and maps

$\mathfrak{e}_{6}\;\text{compact}$
e₆ compact — real Lie algebra
Actual real Lie algebra
$\mathfrak{e}_{6}(\mathbb C)$
e₆(C) — complex Lie algebra
Complexification
$\mathrm{E}_{6}\;\text{compact, simply connected}$
Simply connected compact exceptional group of type E₆
Curated global form
$\mathrm{E}_{6}\;\text{compact, adjoint}$
Adjoint compact exceptional group of type E₆
Curated global form

Claims and sources

Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.

Definition

The compact real form of the complex simple Lie algebra e₆(C).

Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source

real-form:e-6-compact-r#definition

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Real dimension

78

Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source

real-form:e-6-compact-r#dimension

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Complexification

e₆(C)

Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source

real-form:e-6-compact-r#complexification

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Compact real form

Yes

Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source

real-form:e-6-compact-r#compact

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Real rank

0

Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source

real-form:e-6-compact-r#real-rank

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One algebra, several integrations

A real form is algebraic data over R. The linked group records fix particular connected integrations, and state their global topology separately. Compactness of this real algebra should not be used to infer a particular quotient kernel.

Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

real-form:e-6-compact-r#global-forms-1

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Actual real Lie algebra

algebra:e-6-compact-r

Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source

real-form:e-6-compact-r#relation-1

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Complexification

algebra:e-6-c

Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source

real-form:e-6-compact-r#relation-2

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Curated global form

group:e-6-compact-sc

Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

real-form:e-6-compact-r#relation-3

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Curated global form

group:e-6-compact-ad

Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

real-form:e-6-compact-r#relation-4

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