Lie theory · Group

$\mathrm{PSU}(4)$ Adjoint compact group of type A₃

SU(4)/C₄. The quotient of SU(4) by its full scalar center C₄.

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Reference coverageCompact Lie groupSimple algebraGlobal form

Definition

The quotient of SU(4) by its full scalar center C₄.

Definition sources

Real dimension: $15$
Compact rank: $3$
Center: $1$
Fundamental group π₁: $\mathrm{C}_{4}$

Reference properties

Property	Value	Evidence
Real dimension	$15$	Sources
Compact rank	$3$	Sources
Center	$1$	Sources
Fundamental group π₁	$\mathrm{C}_{4}$	Sources
Compact	Yes	Sources
Connected	Yes	Sources
Abelian	No	Sources
Simple real Lie algebra	Yes	Sources
Global form	$\mathrm{SU}(4)/\mathrm{C}_{4}$	Sources
Real Lie algebra	$\mathfrak{su}(4) \cong \mathfrak{so}(6)$	Sources

Global structure and topology

Its simply connected cover is SU(4), with covering kernel C₄. Through Spin(6) ≅ SU(4), this quotient is PSO(6) = SO(6)/{±I}. [sources]

Representation category and descent

An SU(4) highest weight with Dynkin labels (a,b,c) descends exactly when a + 2b + 3c is divisible by 4. This criterion concerns the center action, not the reality type. [sources]

Names and aliases

SU(4)/C4 · PSO(6)

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

Related objects and maps

$\mathfrak{su}(4) \cong \mathfrak{so}(6)$
su(4) ≅ so(6) — real Lie algebra
Real Lie algebra
$\mathfrak{su}(4) \cong \mathfrak{so}(6)$
su(4) ≅ so(6) real form
Real-form context
$\mathrm{SU}(4) \longrightarrow \mathrm{PSU}(4)$
Full central quotient of SU(4)
Explicit homomorphism
$\mathrm{SU}(4)$
Special unitary group in four dimensions
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 SU(4) by its full scalar center C₄.

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:psu-4#definition

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Real dimension

15

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:psu-4#dimension

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Compact rank

3

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

group:psu-4#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:psu-4#center

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Fundamental group π₁

C₄

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:psu-4#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:psu-4#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:psu-4#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:psu-4#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:psu-4#simple-algebra

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Global form

SU(4)/C₄

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
Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

group:psu-4#global-form

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Real Lie algebra

su(4) ≅ so(6)

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:psu-4#lie-algebra

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Global structure and topology

Its simply connected cover is SU(4), with covering kernel C₄. Through Spin(6) ≅ SU(4), this quotient is PSO(6) = SO(6)/{±I}.

Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

group:psu-4#topology-1

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Representation category and descent

An SU(4) highest weight with Dynkin labels (a,b,c) descends exactly when a + 2b + 3c is divisible by 4. This criterion concerns the center action, not the reality type.

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

Derivation: The k-th exterior-power fundamental representation has scalar-center charge k; add charges with the Dynkin-label multiplicities.

Premises: The center generator iI of SU(4) · Fundamental weights realized by Λ¹, Λ², and Λ³ of C⁴

group:psu-4#representations-1

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Real Lie algebra

algebra:su-4-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:psu-4#relation-1

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Real-form context

real-form:su-4-r

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

group:psu-4#relation-2

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Explicit homomorphism

map:su-4-to-psu-4

Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

group:psu-4#relation-3

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Specified connected covering group

group:su-4

Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

group:psu-4#relation-4

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