$\mathrm{SU}(3)$Special unitary group

3 × 3 complex unitary matrices of determinant 1

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source

8

• Lie Groups and Lie AlgebrasSections 6.1–6.3: su(3) has real dimension 3²−1=8Open source

ℤ₃

• Math 210C. Centers of simply connected semisimple groupsPage 3: type A₂ center μ₃, realized by scalar matrices in SU(3)Open source

0

• Conrad and Landesman · Compact Lie groupsLecture 16, printed page 73: SU(n) and Sp(n) are simply connectedOpen source

Yes

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source

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

2

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

An irrep (p,q) has dimension (p+1)(q+1)(p+q+2)/2 and center charge p+2q modulo 3.

• Lie Groups and Lie AlgebrasSection 26.5: Weyl dimension formula, specialized to the three positive A₂ rootsOpen source
• Lie Groups and Lie AlgebrasSection 27.2: SL(n,C) fundamental representations as exterior powersOpen 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

SU(3) is simply connected; its center is generated by exp(2πi/3)I. The adjoint form is PSU(3).

• 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
• Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source

Color symmetry, multiplet decompositions, weight diagrams, and global gauge structure.

• Aharony, Seiberg and Tachikawa · Global gauge structureApplications to the indicated symmetry and representation categoryOpen source

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.

• Lie Groups and Lie AlgebrasSection 26.5: Weyl dimension formula, specialized to the three positive A₂ rootsOpen source
• Lie Groups and Lie AlgebrasSection 27.2: SL(n,C) fundamental representations as exterior powersOpen 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

The Weyl dimension product over α₁, α₂, and α₁+α₂ gives dim(p,q) = (p+1)(q+1)(p+q+2)/2.

• Lie Groups and Lie AlgebrasSection 26.5: Weyl dimension formula, specialized to the three positive A₂ rootsOpen source
• Lie Groups and Lie AlgebrasSection 27.2: SL(n,C) fundamental representations as exterior powersOpen 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

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source

group:su-2

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source

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

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

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

map:su-2-to-su-3-regular

• Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: block SU(2) subgroup of SU(3)Open source

map:su-2-to-su-3-principal

• Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) imageOpen source