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    "released": "2026-11-08"
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  "entity": {
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    "slug": "su-4",
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    "level": "executable",
    "aliases": [
      "Spin(6)"
    ],
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    ],
    "tags": [
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        "source": "etingof-lie-notes",
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        "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
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        "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
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            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
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            "source": "etingof-coxeter",
            "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers"
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            "source": "conrad",
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            "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
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            "source": "conrad",
            "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
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            "source": "conrad",
            "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
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            "source": "conrad",
            "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
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            "source": "conrad",
            "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
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            "source": "conrad",
            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
          }
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            "source": "conrad",
            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
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            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
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            "source": "conrad-centers",
            "locator": "Pages 2–4: centers of simply connected groups by root-system type"
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            "source": "conrad-centers",
            "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
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            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
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          {
            "text": "SU(4) is simply connected. Its center consists of the scalar fourth roots of unity. The identification Spin(6) ≅ SU(4) gives the vector cover SU(4) → SO(6) with kernel {±I}.",
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                "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
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                "source": "conrad",
                "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
              }
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            "assumptions": []
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            "text": "Finite-dimensional complex irreducibles have A₃ dominant integral highest weights. The defining 4 and its dual are the two chiral spin representations in the Spin(6) identification; the exterior square of the 4 is the vector 6.",
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                "locator": "Lectures 27–28: highest-weight classification and fundamental representations"
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                "locator": "Section 2.9: SU(4) acting on the real form of its exterior square"
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    "relationships": [
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            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
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            "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
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            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
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            "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
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            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
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          "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
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          "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
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          "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers"
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      "revision": "2026.11-workbench.4"
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        {
          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
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      "revision": "2026.11-workbench.4"
    },
    {
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          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
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          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
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      "revision": "2026.11-workbench.4"
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        },
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          "source": "conrad",
          "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
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      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
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      "subject": "group:su-4",
      "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-4#abelian",
      "subject": "group:su-4",
      "key": "abelian",
      "label": "Abelian",
      "value": "No",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "conrad",
          "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
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      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
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    {
      "id": "group:su-4#simple-algebra",
      "subject": "group:su-4",
      "key": "simple-algebra",
      "label": "Simple real Lie algebra",
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      "evidence": [
        {
          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
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          "source": "conrad",
          "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
        }
      ],
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      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:su-4#global-form",
      "subject": "group:su-4",
      "key": "global-form",
      "label": "Global form",
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      "evidence": [
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          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
        },
        {
          "source": "conrad-centers",
          "locator": "Pages 2–4: centers of simply connected groups by root-system type"
        },
        {
          "source": "conrad-centers",
          "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
        },
        {
          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
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      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
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      "subject": "group:su-4",
      "key": "lie-algebra",
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          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
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          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
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      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
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      "value": "SU(4) is simply connected. Its center consists of the scalar fourth roots of unity. The identification Spin(6) ≅ SU(4) gives the vector cover SU(4) → SO(6) with kernel {±I}.",
      "evidence": [
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          "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
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          "locator": "Section 2.9: SU(4) acting on the real form of its exterior square"
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          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
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      "relationship": "has-algebra",
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          "source": "conrad-centers",
          "locator": "Pages 2–4: centers of simply connected groups by root-system type"
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          "source": "conrad-centers",
          "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
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          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
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          "source": "conrad-centers",
          "locator": "Pages 2–4: centers of simply connected groups by root-system type"
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        {
          "source": "conrad-centers",
          "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
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        {
          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
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      "title": "Lie Groups and Lie Algebras",
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      "title": "Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual Representations",
      "authors": [
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      "title": "Sporadic isogenies to orthogonal groups",
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