Lie theory · Group

$\mathrm{E}_{8}\;\text{compact}$ Compact exceptional group of type E₈

Simply connected and adjoint compact form of E₈. The connected compact Lie group with compact real Lie algebra of type E₈; its simply connected and adjoint forms agree.

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Executable coverageCompact Lie groupSimple algebraSimply connected

Definition

The connected compact Lie group with compact real Lie algebra of type E₈; its simply connected and adjoint forms agree.

Definition sources

Real dimension: $248$
Compact rank: $8$
Center: $1$
Fundamental group π₁: $0$

Reference properties

Property	Value	Evidence
Real dimension	$248$	Sources
Compact rank	$8$	Sources
Center	$1$	Sources
Fundamental group π₁	$0$	Sources
Compact	Yes	Sources
Connected	Yes	Sources
Abelian	No	Sources
Simple real Lie algebra	Yes	Sources
Global form	Simply connected and adjoint compact form of E₈	Sources
Real Lie algebra	$\mathfrak{e}_{8}\;\text{compact}$	Sources

Global structure and topology

Its center and fundamental group are both trivial. There is no nontrivial connected central quotient of the simply connected compact E₈ group. [sources]

Representation category and descent

The E₈ weight lattice equals its root lattice. Thus every dominant integral highest weight integrates to this single compact global form; its adjoint representation has dimension 248. [sources]

Names and aliases

Compact E8

Aliases identify the stated object under this record’s conventions. Type names and broader search terms are kept separate.

Related objects and maps

$\mathfrak{e}_{8}\;\text{compact}$
e₈ compact — real Lie algebra
Real Lie algebra
$\mathfrak{e}_{8}\;\text{compact}$
e₈ compact real form
Real-form context

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 connected compact Lie group with compact real Lie algebra of type E₈; its simply connected and adjoint forms agree.

Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
Conrad and Landesman · Compact Lie groupsLectures 26–28 and Appendix V: compact semisimple groups and highest weightsOpen source
Conrad and Landesman · Compact Lie groupsLecture 16, printed page 73: SU(n) and Sp(n) are simply connectedOpen source

group:e-8-compact#definition

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Real dimension

248

Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual RepresentationsSection 32.4, printed pages 175–176: Coxeter and dual Coxeter numbersOpen source

group:e-8-compact#dimension

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Compact rank

8

Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source

group:e-8-compact#rank

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Center

1

Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

group:e-8-compact#center

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Fundamental group π₁

0

Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source
Conrad and Landesman · Compact Lie groupsLecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3Open source

group:e-8-compact#pi1

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Compact

Yes

Conrad and Landesman · Compact Lie groupsLectures 26–28 and Appendix V: compact semisimple groups and highest weightsOpen source
Conrad and Landesman · Compact Lie groupsLecture 16, printed page 73: SU(n) and Sp(n) are simply connectedOpen source

group:e-8-compact#compact

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Connected

Yes

Conrad and Landesman · Compact Lie groupsLectures 26–28 and Appendix V: compact semisimple groups and highest weightsOpen source
Conrad and Landesman · Compact Lie groupsLecture 16, printed page 73: SU(n) and Sp(n) are simply connectedOpen source

group:e-8-compact#connected

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Abelian

No

Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source

group:e-8-compact#abelian

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Simple real Lie algebra

Yes

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

group:e-8-compact#simple-algebra

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Global form

Simply connected and adjoint compact form of E₈

Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source
Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

group:e-8-compact#global-form

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Real Lie algebra

e₈ compact

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

group:e-8-compact#lie-algebra

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Global structure and topology

Its center and fundamental group are both trivial. There is no nontrivial connected central quotient of the simply connected compact E₈ group.

Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

group:e-8-compact#topology-1

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Representation category and descent

The E₈ weight lattice equals its root lattice. Thus every dominant integral highest weight integrates to this single compact global form; its adjoint representation has dimension 248.

Conrad and Landesman · Compact Lie groupsLectures 27–28: highest-weight classification and fundamental representationsOpen source
Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotientsOpen source
Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source

group:e-8-compact#representations-1

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Real Lie algebra

algebra:e-8-compact-r

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

group:e-8-compact#relation-1

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Real-form context

real-form:e-8-compact-r

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

group:e-8-compact#relation-2

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