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  "versions": {
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    "dataset": "2026.11-workbench.4",
    "application": "6.0.0",
    "algorithm": "groups-workbench-4.0.0",
    "released": "2026-11-08"
  },
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  "entity": {
    "id": "map:su-2-to-su-3-regular",
    "slug": "su-2-to-su-3-regular",
    "name": "Regular block inclusion in SU(3)",
    "symbol": "SU(2) ↪ SU(3)",
    "category": "lie",
    "level": "executable",
    "aliases": [],
    "searchTerms": [],
    "tags": [
      "Embedding"
    ],
    "summary": "A ↦ diag(A,1). The first two coordinates form the defining SU(2) doublet and the third coordinate is fixed. Kernel: {I}.",
    "definition": "A ↦ diag(A,1). The first two coordinates form the defining SU(2) doublet and the third coordinate is fixed.",
    "definitionEvidence": [
      {
        "source": "haber",
        "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
      }
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        "key": "domain",
        "label": "Domain",
        "value": "su-2",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
          }
        ],
        "assumptions": []
      },
      {
        "key": "codomain",
        "label": "Codomain",
        "value": "su-3",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
          }
        ],
        "assumptions": []
      },
      {
        "key": "formula",
        "label": "Specified homomorphism",
        "value": "A ↦ diag(A,1). The first two coordinates form the defining SU(2) doublet and the third coordinate is fixed.",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
          }
        ],
        "assumptions": []
      },
      {
        "key": "kernel",
        "label": "Kernel",
        "value": "{I}",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
          }
        ],
        "assumptions": []
      },
      {
        "key": "image",
        "label": "Image",
        "value": "{diag(A,1) : A ∈ SU(2)} ≅ SU(2)",
        "evidence": [
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            "source": "haber",
            "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
          }
        ],
        "assumptions": []
      },
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        "key": "injective",
        "label": "Injective",
        "value": "Yes",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
          }
        ],
        "assumptions": []
      },
      {
        "key": "surjective",
        "label": "Surjective",
        "value": "No",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
          }
        ],
        "assumptions": []
      }
    ],
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        "id": "meaning",
        "title": "Global and representation consequences",
        "paragraphs": [
          {
            "text": "The SU(3) fundamental branches as 3 → 2 ⊕ 1. The adjoint branches as 8 → 3 ⊕ 2 ⊕ 2 ⊕ 1.",
            "evidence": [
              {
                "source": "haber",
                "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
              },
              {
                "source": "etingof-global-forms",
                "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
              }
            ],
            "assumptions": []
          },
          {
            "text": "For the adjoint rule, restrict End(C³) and remove its scalar summand. End(2 ⊕ 1) decomposes as (3 ⊕ 1) ⊕ 2 ⊕ 2 ⊕ 1, giving the stated eight-dimensional result after subtracting one singlet.",
            "evidence": [
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                "source": "conrad",
                "locator": "Lectures 27–28: highest-weight classification and fundamental representations"
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              {
                "source": "haber",
                "locator": "Problem 5: regular inclusion"
              }
            ],
            "derivation": {
              "method": "Restrict End(3) − 1 using the specified block inclusion and the SU(2) tensor rule 2⊗2=3⊕1.",
              "inputs": [
                "3 restricts to 2⊕1",
                "Adjoint SU(3) = End(3) minus scalars"
              ]
            },
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "has-domain",
        "target": "group:su-2",
        "label": "Domain group",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
          }
        ]
      },
      {
        "type": "has-codomain",
        "target": "group:su-3",
        "label": "Codomain group",
        "evidence": [
          {
            "source": "haber",
            "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
          }
        ]
      }
    ],
    "capabilities": [
      "branching",
      "su3-branching"
    ],
    "kind": "map",
    "data": {
      "domain": "group:su-2",
      "codomain": "group:su-3",
      "kernel": "{I}",
      "image": "{diag(A,1) : A ∈ SU(2)} ≅ SU(2)",
      "injective": true,
      "surjective": false,
      "capabilityPresets": {
        "branching": {
          "embedding": "regular",
          "rep": "8"
        },
        "su3-branching": {
          "labels": "1,1",
          "map": "regular"
        }
      }
    }
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  "claims": [
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      "subject": "map:su-2-to-su-3-regular",
      "key": "definition",
      "label": "Definition",
      "value": "A ↦ diag(A,1). The first two coordinates form the defining SU(2) doublet and the third coordinate is fixed.",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
        }
      ],
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      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-to-su-3-regular#domain",
      "subject": "map:su-2-to-su-3-regular",
      "key": "domain",
      "label": "Domain",
      "value": "su-2",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
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      "subject": "map:su-2-to-su-3-regular",
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          "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
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      "assumptions": [],
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    {
      "id": "map:su-2-to-su-3-regular#formula",
      "subject": "map:su-2-to-su-3-regular",
      "key": "formula",
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      "value": "A ↦ diag(A,1). The first two coordinates form the defining SU(2) doublet and the third coordinate is fixed.",
      "evidence": [
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          "source": "haber",
          "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
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      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-to-su-3-regular#kernel",
      "subject": "map:su-2-to-su-3-regular",
      "key": "kernel",
      "label": "Kernel",
      "value": "{I}",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
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      "id": "map:su-2-to-su-3-regular#image",
      "subject": "map:su-2-to-su-3-regular",
      "key": "image",
      "label": "Image",
      "value": "{diag(A,1) : A ∈ SU(2)} ≅ SU(2)",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
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      "id": "map:su-2-to-su-3-regular#injective",
      "subject": "map:su-2-to-su-3-regular",
      "key": "injective",
      "label": "Injective",
      "value": "Yes",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-to-su-3-regular#surjective",
      "subject": "map:su-2-to-su-3-regular",
      "key": "surjective",
      "label": "Surjective",
      "value": "No",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-to-su-3-regular#meaning-1",
      "subject": "map:su-2-to-su-3-regular",
      "key": "meaning-1",
      "label": "Global and representation consequences",
      "value": "The SU(3) fundamental branches as 3 → 2 ⊕ 1. The adjoint branches as 8 → 3 ⊕ 2 ⊕ 2 ⊕ 1.",
      "evidence": [
        {
          "source": "haber",
          "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
        },
        {
          "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-regular#meaning-2",
      "subject": "map:su-2-to-su-3-regular",
      "key": "meaning-2",
      "label": "Global and representation consequences",
      "value": "For the adjoint rule, restrict End(C³) and remove its scalar summand. End(2 ⊕ 1) decomposes as (3 ⊕ 1) ⊕ 2 ⊕ 2 ⊕ 1, giving the stated eight-dimensional result after subtracting one singlet.",
      "evidence": [
        {
          "source": "conrad",
          "locator": "Lectures 27–28: highest-weight classification and fundamental representations"
        },
        {
          "source": "haber",
          "locator": "Problem 5: regular inclusion"
        }
      ],
      "derivation": {
        "method": "Restrict End(3) − 1 using the specified block inclusion and the SU(2) tensor rule 2⊗2=3⊕1.",
        "inputs": [
          "3 restricts to 2⊕1",
          "Adjoint SU(3) = End(3) minus scalars"
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
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      "id": "map:su-2-to-su-3-regular#relation-1",
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          "source": "haber",
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      "revision": "2026.11-workbench.4"
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      "key": "relation-2",
      "label": "Codomain group",
      "value": "group:su-3",
      "evidence": [
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          "source": "haber",
          "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)"
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      ],
      "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"
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      "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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