Projective special unitary group
The adjoint compact group SU(3)/ℤ₃. It shares its Lie algebra with SU(3), with a different global form.
Reference properties
Representations
An SU(3) irrep (p,q) descends precisely when p+2q is zero modulo 3. Descent does not imply real type. [sources]
Topology and global form
The covering SU(3) → PSU(3) has kernel ℤ₃, which is the fundamental group of PSU(3). [sources]
In physics
Allowed matter, Wilson lines, and the distinction between local and global gauge structure. [sources]
A₂ Dynkin-label convention
The labels (p,q) mean λ = pω₁ + qω₂, with (1,0) the defining 3 and (0,1) its conjugate. The scalar center generator exp(2πi/3)I acts with charge p + 2q modulo 3. [sources]
Names and aliases
SU(3)/Z3
Related objects and maps
- Related group
- Actual real Lie algebra
- Real-form context
- Explicit homomorphism
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
SU(3) modulo scalar cube roots of unity
- 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-3#definitionReal dimension
8
- Lie Groups and Lie AlgebrasSections 6.1–6.3: dim su(3)=8; a discrete central quotient preserves the Lie algebraOpen source
group:psu-3#dimensionCenter
Trivial
- Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.1: the compact adjoint form has trivial centerOpen source
group:psu-3#centerFundamental group
ℤ₃
- 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-3#pi1Compact
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-3#compactConnected
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-3#connectedRank
2
- Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
group:psu-3#rankRepresentations
An SU(3) irrep (p,q) descends precisely when p+2q is zero modulo 3. Descent does not imply real type.
- Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.1, Proposition 43.2: adjoint-group weights are the root latticeOpen source
- Lie Groups and Lie AlgebrasSection 27.2: the two A₂ fundamental modules have scalar charges 1 and 2Open source
group:psu-3#representations-1Topology and global form
The covering SU(3) → PSU(3) has kernel ℤ₃, which is the fundamental group of PSU(3).
- 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-3#topology-1In physics
Allowed matter, Wilson lines, and the distinction between local and global gauge structure.
- Aharony, Seiberg and Tachikawa · Global gauge structureApplications to the indicated symmetry and representation categoryOpen source
group:psu-3#physics-1A₂ Dynkin-label convention
The labels (p,q) mean λ = pω₁ + qω₂, with (1,0) the defining 3 and (0,1) its conjugate. The scalar center generator exp(2πi/3)I acts with charge p + 2q modulo 3.
- Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.1, Proposition 43.2: adjoint-group weights are the root latticeOpen source
- Lie Groups and Lie AlgebrasSection 27.2: the two A₂ fundamental modules have scalar charges 1 and 2Open source
Premises: Fundamental modules C³ and Λ²C³ · Dynkin labels p,q
group:psu-3#weight-convention-1Related group
group:su-3
- 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-3#relation-1Actual real Lie algebra
algebra:su-3-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-3#relation-2Real-form context
real-form:su-3-r
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
group:psu-3#relation-3Explicit homomorphism
map:su-3-to-psu-3
- 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
- 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:psu-3#relation-4