{
  "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/maps/su-2-to-su-3-principal/",
  "entity": {
    "id": "map:su-2-to-su-3-principal",
    "slug": "su-2-to-su-3-principal",
    "name": "Principal spin-one homomorphism into SU(3)",
    "symbol": "SU(2) → SU(3), spin 1",
    "category": "lie",
    "level": "executable",
    "aliases": [],
    "searchTerms": [],
    "tags": [
      "Noninjective homomorphism"
    ],
    "summary": "A ↦ Sym²(A) on Sym²(C²), with a unitary basis fixed. A real orthonormal form identifies its image with SO(3). Kernel: {±I} ≅ C₂.",
    "definition": "A ↦ Sym²(A) on Sym²(C²), with a unitary basis fixed. A real orthonormal form identifies its image with SO(3).",
    "definitionEvidence": [
      {
        "source": "haber",
        "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
      }
    ],
    "properties": [
      {
        "key": "domain",
        "label": "Domain",
        "value": "su-2",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
          }
        ],
        "assumptions": []
      },
      {
        "key": "codomain",
        "label": "Codomain",
        "value": "su-3",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
          }
        ],
        "assumptions": []
      },
      {
        "key": "formula",
        "label": "Specified homomorphism",
        "value": "A ↦ Sym²(A) on Sym²(C²), with a unitary basis fixed. A real orthonormal form identifies its image with SO(3).",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
          }
        ],
        "assumptions": []
      },
      {
        "key": "kernel",
        "label": "Kernel",
        "value": "{±I} ≅ C₂",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
          }
        ],
        "assumptions": []
      },
      {
        "key": "image",
        "label": "Image",
        "value": "An SU(3)-conjugate of the real SO(3) subgroup",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
          }
        ],
        "assumptions": []
      },
      {
        "key": "injective",
        "label": "Injective",
        "value": "No",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
          }
        ],
        "assumptions": []
      },
      {
        "key": "surjective",
        "label": "Surjective",
        "value": "No",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
          }
        ],
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "meaning",
        "title": "Global and representation consequences",
        "paragraphs": [
          {
            "text": "The fundamental branches as 3 → 3, and the adjoint as 8 → 3 ⊕ 5. This is a Lie-algebra embedding, but the group homomorphism is not injective.",
            "evidence": [
              {
                "source": "haber",
                "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
              },
              {
                "source": "etingof-global-forms",
                "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
              }
            ],
            "assumptions": []
          },
          {
            "text": "The adjoint rule follows from 3 ⊗ 3 = 1 ⊕ 3 ⊕ 5 after removing the scalar endomorphism. Every surviving summand has integer spin, consistent with the central kernel.",
            "evidence": [
              {
                "source": "conrad",
                "locator": "Lectures 27–28: highest-weight classification and fundamental representations"
              }
            ],
            "derivation": {
              "method": "Use the SU(2) spin-one tensor decomposition and Adj(SU(3)) = End(3) − 1.",
              "inputs": [
                "The defining SU(3) representation restricts to spin 1",
                "Spin 1 ⊗ spin 1 = spin 0 ⊕ spin 1 ⊕ spin 2"
              ]
            },
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "has-domain",
        "target": "group:su-2",
        "label": "Domain group",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
          }
        ]
      },
      {
        "type": "has-codomain",
        "target": "group:su-3",
        "label": "Codomain group",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
          }
        ]
      }
    ],
    "capabilities": [
      "branching",
      "su3-branching"
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    "kind": "map",
    "data": {
      "domain": "group:su-2",
      "codomain": "group:su-3",
      "kernel": "{±I} ≅ C₂",
      "image": "An SU(3)-conjugate of the real SO(3) subgroup",
      "injective": false,
      "surjective": false,
      "capabilityPresets": {
        "branching": {
          "embedding": "principal",
          "rep": "8"
        },
        "su3-branching": {
          "labels": "1,1",
          "map": "principal"
        }
      }
    }
  },
  "claims": [
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      "subject": "map:su-2-to-su-3-principal",
      "key": "definition",
      "label": "Definition",
      "value": "A ↦ Sym²(A) on Sym²(C²), with a unitary basis fixed. A real orthonormal form identifies its image with SO(3).",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
        }
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      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-to-su-3-principal#domain",
      "subject": "map:su-2-to-su-3-principal",
      "key": "domain",
      "label": "Domain",
      "value": "su-2",
      "evidence": [
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          "source": "haber",
          "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-to-su-3-principal#codomain",
      "subject": "map:su-2-to-su-3-principal",
      "key": "codomain",
      "label": "Codomain",
      "value": "su-3",
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          "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
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    },
    {
      "id": "map:su-2-to-su-3-principal#formula",
      "subject": "map:su-2-to-su-3-principal",
      "key": "formula",
      "label": "Specified homomorphism",
      "value": "A ↦ Sym²(A) on Sym²(C²), with a unitary basis fixed. A real orthonormal form identifies its image with SO(3).",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-to-su-3-principal#kernel",
      "subject": "map:su-2-to-su-3-principal",
      "key": "kernel",
      "label": "Kernel",
      "value": "{±I} ≅ C₂",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-to-su-3-principal#image",
      "subject": "map:su-2-to-su-3-principal",
      "key": "image",
      "label": "Image",
      "value": "An SU(3)-conjugate of the real SO(3) subgroup",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-to-su-3-principal#injective",
      "subject": "map:su-2-to-su-3-principal",
      "key": "injective",
      "label": "Injective",
      "value": "No",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-to-su-3-principal#surjective",
      "subject": "map:su-2-to-su-3-principal",
      "key": "surjective",
      "label": "Surjective",
      "value": "No",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-to-su-3-principal#meaning-1",
      "subject": "map:su-2-to-su-3-principal",
      "key": "meaning-1",
      "label": "Global and representation consequences",
      "value": "The fundamental branches as 3 → 3, and the adjoint as 8 → 3 ⊕ 5. This is a Lie-algebra embedding, but the group homomorphism is not injective.",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
        },
        {
          "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": "map:su-2-to-su-3-principal#meaning-2",
      "subject": "map:su-2-to-su-3-principal",
      "key": "meaning-2",
      "label": "Global and representation consequences",
      "value": "The adjoint rule follows from 3 ⊗ 3 = 1 ⊕ 3 ⊕ 5 after removing the scalar endomorphism. Every surviving summand has integer spin, consistent with the central kernel.",
      "evidence": [
        {
          "source": "conrad",
          "locator": "Lectures 27–28: highest-weight classification and fundamental representations"
        }
      ],
      "derivation": {
        "method": "Use the SU(2) spin-one tensor decomposition and Adj(SU(3)) = End(3) − 1.",
        "inputs": [
          "The defining SU(3) representation restricts to spin 1",
          "Spin 1 ⊗ spin 1 = spin 0 ⊕ spin 1 ⊕ spin 2"
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-to-su-3-principal#relation-1",
      "subject": "map:su-2-to-su-3-principal",
      "key": "relation-1",
      "label": "Domain group",
      "value": "group:su-2",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
        }
      ],
      "assumptions": [],
      "relationship": "has-domain",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-to-su-3-principal#relation-2",
      "subject": "map:su-2-to-su-3-principal",
      "key": "relation-2",
      "label": "Codomain group",
      "value": "group:su-3",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image"
        }
      ],
      "assumptions": [],
      "relationship": "has-codomain",
      "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",
        "Aaron Landesman"
      ],
      "url": "https://math.stanford.edu/~conrad/210CPage/handouts/lie_groups_notes.pdf",
      "kind": "lecture-notes",
      "locatorHint": "Numbered lectures and appendices; page numbers refer to the printed pagination.",
      "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted."
    },
    {
      "id": "etingof-global-forms",
      "title": "Lecture 17: Classification of Connected Compact and Complex Reductive Groups",
      "authors": [
        "Pavel Etingof"
      ],
      "url": "https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec17.pdf",
      "kind": "lecture-notes",
      "locatorHint": "Sections 43.1–43.2, printed pages 230–235.",
      "licenseNote": "Linked MIT OpenCourseWare notes; global-form and representation criteria are restated."
    }
  ],
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