Lie theory · Group

$\mathrm{E}_{6}\;\text{compact, adjoint}$ Adjoint compact exceptional group of type E₆

E₆ simply connected / C₃. The quotient of the simply connected compact E₆ group by its full center C₃.

Compare this group

Reference coverageCompact Lie groupSimple algebraGlobal form

Definition

The quotient of the simply connected compact E₆ group by its full center C₃.

Definition sources

Real dimension: $78$
Compact rank: $6$
Center: $1$
Fundamental group π₁: $\mathrm{C}_{3}$

Reference properties

Property	Value	Evidence
Real dimension	$78$	Sources
Compact rank	$6$	Sources
Center	$1$	Sources
Fundamental group π₁	$\mathrm{C}_{3}$	Sources
Compact	Yes	Sources
Connected	Yes	Sources
Abelian	No	Sources
Simple real Lie algebra	Yes	Sources
Global form	E₆ simply connected / C₃	Sources
Real Lie algebra	$\mathfrak{e}_{6}\;\text{compact}$	Sources

Global structure and topology

The adjoint group has trivial center and fundamental group C₃. The cover has the same local real algebra and dimension. [sources]

Representation category and descent

Finite-dimensional complex irreducibles are the dominant E₆ weights lying in the root lattice. This is the zero-center-character sector of the simply connected group. [sources]

Names and aliases

E6/C3 · E6/Z3

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

Related objects and maps

$\mathfrak{e}_{6}\;\text{compact}$
e₆ compact — real Lie algebra
Real Lie algebra
$\mathfrak{e}_{6}\;\text{compact}$
e₆ compact real form
Real-form context
$\mathrm{E}_{6}\;\text{compact, simply connected}$
Simply connected compact exceptional group of type E₆
Specified connected covering group

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 quotient of the simply connected compact E₆ group by its full center C₃.

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-6-compact-ad#definition

Link to claim Suggest a correction

Real dimension

78

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-6-compact-ad#dimension

Link to claim Suggest a correction

Compact rank

6

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

group:e-6-compact-ad#rank

Link to claim Suggest a correction

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-6-compact-ad#center

Link to claim Suggest a correction

Fundamental group π₁

C₃

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-6-compact-ad#pi1

Link to claim Suggest a correction

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-6-compact-ad#compact

Link to claim Suggest a correction

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-6-compact-ad#connected

Link to claim Suggest a correction

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-6-compact-ad#abelian

Link to claim Suggest a correction

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-6-compact-ad#simple-algebra

Link to claim Suggest a correction

Global form

E₆ simply connected / C₃

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-6-compact-ad#global-form

Link to claim Suggest a correction

Real Lie algebra

e₆ 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-6-compact-ad#lie-algebra

Link to claim Suggest a correction

Global structure and topology

The adjoint group has trivial center and fundamental group C₃. The cover has the same local real algebra and dimension.

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-6-compact-ad#topology-1

Link to claim Suggest a correction

Representation category and descent

Finite-dimensional complex irreducibles are the dominant E₆ weights lying in the root lattice. This is the zero-center-character sector of the simply connected group.

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

group:e-6-compact-ad#representations-1

Link to claim Suggest a correction

Real Lie algebra

algebra:e-6-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-6-compact-ad#relation-1

Link to claim Suggest a correction

Real-form context

real-form: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

group:e-6-compact-ad#relation-2

Link to claim Suggest a correction

Specified connected covering group

group:e-6-compact-sc

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-6-compact-ad#relation-3

Link to claim Suggest a correction