{
  "schema": "groups.qft.org/entity/2",
  "versions": {
    "schema": "2.0.0",
    "dataset": "2026.11-workbench.4",
    "application": "6.0.0",
    "algorithm": "groups-workbench-4.0.0",
    "released": "2026-11-08"
  },
  "canonicalURL": "https://groups.qft.org/atlas/groups/su-3/",
  "entity": {
    "id": "group:su-3",
    "slug": "su-3",
    "name": "Special unitary group",
    "symbol": "SU(3)",
    "category": "lie",
    "level": "executable",
    "aliases": [],
    "searchTerms": [
      "A2"
    ],
    "tags": [
      "Gauge theory",
      "Color"
    ],
    "summary": "The simply connected compact group of type A₂. Highest weights have two nonnegative Dynkin labels (p, q).",
    "definition": "3 × 3 complex unitary matrices of determinant 1",
    "definitionEvidence": [
      {
        "source": "etingof-lie-notes",
        "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
      }
    ],
    "properties": [
      {
        "key": "dimension",
        "label": "Real dimension",
        "value": "8",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: su(3) has real dimension 3²−1=8"
          }
        ],
        "assumptions": []
      },
      {
        "key": "center",
        "label": "Center",
        "value": "ℤ₃",
        "evidence": [
          {
            "source": "conrad-centers",
            "locator": "Page 3: type A₂ center μ₃, realized by scalar matrices in SU(3)"
          }
        ],
        "assumptions": []
      },
      {
        "key": "pi1",
        "label": "Fundamental group",
        "value": "0",
        "evidence": [
          {
            "source": "conrad",
            "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
          }
        ],
        "assumptions": []
      },
      {
        "key": "compact",
        "label": "Compact",
        "value": "Yes",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          }
        ],
        "assumptions": []
      },
      {
        "key": "connected",
        "label": "Connected",
        "value": "Yes",
        "evidence": [
          {
            "source": "conrad",
            "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
          },
          {
            "source": "conrad",
            "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
          }
        ],
        "assumptions": []
      },
      {
        "key": "rank",
        "label": "Rank",
        "value": "2",
        "evidence": [
          {
            "source": "conrad",
            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
          }
        ],
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "representations",
        "title": "Representations",
        "paragraphs": [
          {
            "text": "An irrep (p,q) has dimension (p+1)(q+1)(p+q+2)/2 and center charge p+2q modulo 3.",
            "evidence": [
              {
                "source": "etingof-lie-notes",
                "locator": "Section 26.5: Weyl dimension formula, specialized to the three positive A₂ roots"
              },
              {
                "source": "etingof-lie-notes",
                "locator": "Section 27.2: SL(n,C) fundamental representations as exterior powers"
              },
              {
                "source": "etingof-global-forms",
                "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
              }
            ],
            "assumptions": []
          }
        ]
      },
      {
        "id": "topology",
        "title": "Topology and global form",
        "paragraphs": [
          {
            "text": "SU(3) is simply connected; its center is generated by exp(2πi/3)I. The adjoint form is PSU(3).",
            "evidence": [
              {
                "source": "conrad",
                "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
              },
              {
                "source": "conrad",
                "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
              },
              {
                "source": "conrad-centers",
                "locator": "Pages 2–4: centers of simply connected groups by root-system type"
              }
            ],
            "assumptions": []
          }
        ]
      },
      {
        "id": "physics",
        "title": "In physics",
        "paragraphs": [
          {
            "text": "Color symmetry, multiplet decompositions, weight diagrams, and global gauge structure.",
            "evidence": [
              {
                "source": "ast",
                "locator": "Applications to the indicated symmetry and representation category"
              }
            ],
            "assumptions": []
          }
        ]
      },
      {
        "id": "weight-convention",
        "title": "A₂ Dynkin-label convention",
        "paragraphs": [
          {
            "text": "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.",
            "evidence": [
              {
                "source": "etingof-lie-notes",
                "locator": "Section 26.5: Weyl dimension formula, specialized to the three positive A₂ roots"
              },
              {
                "source": "etingof-lie-notes",
                "locator": "Section 27.2: SL(n,C) fundamental representations as exterior powers"
              },
              {
                "source": "etingof-global-forms",
                "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
              }
            ],
            "derivation": {
              "method": "The defining module has charge 1 and its exterior square has charge 2; highest-weight charges add.",
              "inputs": [
                "Fundamental modules C³ and Λ²C³",
                "Dynkin labels p,q"
              ]
            },
            "assumptions": []
          },
          {
            "text": "The Weyl dimension product over α₁, α₂, and α₁+α₂ gives dim(p,q) = (p+1)(q+1)(p+q+2)/2.",
            "evidence": [
              {
                "source": "etingof-lie-notes",
                "locator": "Section 26.5: Weyl dimension formula, specialized to the three positive A₂ roots"
              },
              {
                "source": "etingof-lie-notes",
                "locator": "Section 27.2: SL(n,C) fundamental representations as exterior powers"
              },
              {
                "source": "etingof-global-forms",
                "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
              }
            ],
            "derivation": {
              "method": "Apply the Weyl dimension formula to the three positive roots of A₂.",
              "inputs": [
                "λ=pω₁+qω₂",
                "ρ=ω₁+ω₂",
                "Positive-root coroot pairings with ρ are 1,1,2"
              ]
            },
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "related",
        "target": "group:psu-3",
        "label": "Related group"
      },
      {
        "type": "related",
        "target": "group:su-2",
        "label": "Related group"
      },
      {
        "type": "has-algebra",
        "target": "algebra:su-3-r",
        "label": "Actual real Lie algebra",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          },
          {
            "source": "etingof-real-forms",
            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          }
        ]
      },
      {
        "type": "has-real-form",
        "target": "real-form:su-3-r",
        "label": "Real-form context",
        "evidence": [
          {
            "source": "etingof-real-forms",
            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          }
        ]
      },
      {
        "type": "has-map",
        "target": "map:su-3-to-psu-3",
        "label": "Explicit homomorphism",
        "evidence": [
          {
            "source": "conrad-centers",
            "locator": "Pages 2–4: centers of simply connected groups by root-system type"
          },
          {
            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
          },
          {
            "source": "etingof-global-forms",
            "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
          }
        ]
      },
      {
        "type": "has-map",
        "target": "map:su-2-to-su-3-regular",
        "label": "Explicit homomorphism",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
          }
        ]
      },
      {
        "type": "has-map",
        "target": "map:su-2-to-su-3-principal",
        "label": "Explicit homomorphism",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
          }
        ]
      }
    ],
    "capabilities": [
      "global-forms",
      "branching",
      "lie-representations",
      "matter-quotients",
      "su3-tensor-products",
      "su3-branching",
      "sun-tensor-products",
      "charged-branching"
    ],
    "kind": "group",
    "data": {
      "category": "lie",
      "dimension": 8,
      "compact": true,
      "connected": true,
      "algebraId": "algebra:su-3-r",
      "capabilityPresets": {
        "lie-representations": {
          "type": "A",
          "rank": "2",
          "labels": "1,0"
        },
        "matter-quotients": {
          "n": "3",
          "matter": "2,2",
          "quotient": "1"
        },
        "su3-tensor-products": {
          "left": "1,1",
          "right": "1,1"
        },
        "su3-branching": {
          "labels": "1,1",
          "map": "regular"
        },
        "sun-tensor-products": {
          "n": "3",
          "left": "1,0",
          "right": "0,1"
        },
        "charged-branching": {
          "n": "3",
          "p": "2",
          "labels": "1,0"
        }
      }
    }
  },
  "claims": [
    {
      "id": "group:su-3#definition",
      "subject": "group:su-3",
      "key": "definition",
      "label": "Definition",
      "value": "3 × 3 complex unitary matrices of determinant 1",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#dimension",
      "subject": "group:su-3",
      "key": "dimension",
      "label": "Real dimension",
      "value": "8",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: su(3) has real dimension 3²−1=8"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#center",
      "subject": "group:su-3",
      "key": "center",
      "label": "Center",
      "value": "ℤ₃",
      "evidence": [
        {
          "source": "conrad-centers",
          "locator": "Page 3: type A₂ center μ₃, realized by scalar matrices in SU(3)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#pi1",
      "subject": "group:su-3",
      "key": "pi1",
      "label": "Fundamental group",
      "value": "0",
      "evidence": [
        {
          "source": "conrad",
          "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#compact",
      "subject": "group:su-3",
      "key": "compact",
      "label": "Compact",
      "value": "Yes",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#connected",
      "subject": "group:su-3",
      "key": "connected",
      "label": "Connected",
      "value": "Yes",
      "evidence": [
        {
          "source": "conrad",
          "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
        },
        {
          "source": "conrad",
          "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#rank",
      "subject": "group:su-3",
      "key": "rank",
      "label": "Rank",
      "value": "2",
      "evidence": [
        {
          "source": "conrad",
          "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#representations-1",
      "subject": "group:su-3",
      "key": "representations-1",
      "label": "Representations",
      "value": "An irrep (p,q) has dimension (p+1)(q+1)(p+q+2)/2 and center charge p+2q modulo 3.",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Section 26.5: Weyl dimension formula, specialized to the three positive A₂ roots"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 27.2: SL(n,C) fundamental representations as exterior powers"
        },
        {
          "source": "etingof-global-forms",
          "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#topology-1",
      "subject": "group:su-3",
      "key": "topology-1",
      "label": "Topology and global form",
      "value": "SU(3) is simply connected; its center is generated by exp(2πi/3)I. The adjoint form is PSU(3).",
      "evidence": [
        {
          "source": "conrad",
          "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
        },
        {
          "source": "conrad",
          "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
        },
        {
          "source": "conrad-centers",
          "locator": "Pages 2–4: centers of simply connected groups by root-system type"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#physics-1",
      "subject": "group:su-3",
      "key": "physics-1",
      "label": "In physics",
      "value": "Color symmetry, multiplet decompositions, weight diagrams, and global gauge structure.",
      "evidence": [
        {
          "source": "ast",
          "locator": "Applications to the indicated symmetry and representation category"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#weight-convention-1",
      "subject": "group:su-3",
      "key": "weight-convention-1",
      "label": "A₂ Dynkin-label convention",
      "value": "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.",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Section 26.5: Weyl dimension formula, specialized to the three positive A₂ roots"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 27.2: SL(n,C) fundamental representations as exterior powers"
        },
        {
          "source": "etingof-global-forms",
          "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
        }
      ],
      "derivation": {
        "method": "The defining module has charge 1 and its exterior square has charge 2; highest-weight charges add.",
        "inputs": [
          "Fundamental modules C³ and Λ²C³",
          "Dynkin labels p,q"
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#weight-convention-2",
      "subject": "group:su-3",
      "key": "weight-convention-2",
      "label": "A₂ Dynkin-label convention",
      "value": "The Weyl dimension product over α₁, α₂, and α₁+α₂ gives dim(p,q) = (p+1)(q+1)(p+q+2)/2.",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Section 26.5: Weyl dimension formula, specialized to the three positive A₂ roots"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 27.2: SL(n,C) fundamental representations as exterior powers"
        },
        {
          "source": "etingof-global-forms",
          "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
        }
      ],
      "derivation": {
        "method": "Apply the Weyl dimension formula to the three positive roots of A₂.",
        "inputs": [
          "λ=pω₁+qω₂",
          "ρ=ω₁+ω₂",
          "Positive-root coroot pairings with ρ are 1,1,2"
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#relation-1",
      "subject": "group:su-3",
      "key": "relation-1",
      "label": "Related group",
      "value": "group:psu-3",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        }
      ],
      "assumptions": [],
      "relationship": "related",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#relation-2",
      "subject": "group:su-3",
      "key": "relation-2",
      "label": "Related group",
      "value": "group:su-2",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        }
      ],
      "assumptions": [],
      "relationship": "related",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#relation-3",
      "subject": "group:su-3",
      "key": "relation-3",
      "label": "Actual real Lie algebra",
      "value": "algebra:su-3-r",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
        }
      ],
      "assumptions": [],
      "relationship": "has-algebra",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#relation-4",
      "subject": "group:su-3",
      "key": "relation-4",
      "label": "Real-form context",
      "value": "real-form:su-3-r",
      "evidence": [
        {
          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
        }
      ],
      "assumptions": [],
      "relationship": "has-real-form",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#relation-5",
      "subject": "group:su-3",
      "key": "relation-5",
      "label": "Explicit homomorphism",
      "value": "map:su-3-to-psu-3",
      "evidence": [
        {
          "source": "conrad-centers",
          "locator": "Pages 2–4: centers of simply connected groups by root-system type"
        },
        {
          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
        },
        {
          "source": "etingof-global-forms",
          "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
        }
      ],
      "assumptions": [],
      "relationship": "has-map",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#relation-6",
      "subject": "group:su-3",
      "key": "relation-6",
      "label": "Explicit homomorphism",
      "value": "map:su-2-to-su-3-regular",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
        }
      ],
      "assumptions": [],
      "relationship": "has-map",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-3#relation-7",
      "subject": "group:su-3",
      "key": "relation-7",
      "label": "Explicit homomorphism",
      "value": "map:su-2-to-su-3-principal",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
        }
      ],
      "assumptions": [],
      "relationship": "has-map",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    }
  ],
  "sources": [
    {
      "id": "haber",
      "title": "Haber · Spin covers and subgroup examples",
      "authors": [
        "Howard E. Haber"
      ],
      "url": "https://scipp-legacy.pbsci.ucsc.edu/~haber/archives/physics251_17/groups17_sol4.pdf",
      "kind": "lecture-notes",
      "locatorHint": "Spring 2017; Spin(4) on pages 7–8, SU(2) and SO(3) subgroups of SU(3) on pages 14–15.",
      "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted."
    },
    {
      "id": "conrad",
      "title": "Conrad and Landesman · Compact Lie groups",
      "authors": [
        "Brian Conrad",
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