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    "id": "map:sl-2-c-to-lorentz",
    "slug": "sl-2-c-to-lorentz",
    "name": "Hermitian-matrix Lorentz spin cover",
    "symbol": "SL(2, C) → SO₀(1,3)",
    "category": "lie",
    "level": "reference",
    "aliases": [],
    "searchTerms": [],
    "tags": [
      "Covering or central quotient"
    ],
    "summary": "Identify x with X = x⁰I + x¹σ₁ + x²σ₂ + x³σ₃, a Hermitian 2 × 2 matrix. Send X to AXA† for A ∈ SL(2,C). Then det X = (x⁰)² − (x¹)² − (x²)² − (x³)² is preserved. Kernel: {±I} ≅ C₂.",
    "definition": "Identify x with X = x⁰I + x¹σ₁ + x²σ₂ + x³σ₃, a Hermitian 2 × 2 matrix. Send X to AXA† for A ∈ SL(2,C). Then det X = (x⁰)² − (x¹)² − (x²)² − (x³)² is preserved.",
    "definitionEvidence": [
      {
        "source": "garrett-isogenies",
        "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
      }
    ],
    "properties": [
      {
        "key": "domain",
        "label": "Domain",
        "value": "sl-2-c",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
          }
        ],
        "assumptions": []
      },
      {
        "key": "codomain",
        "label": "Codomain",
        "value": "so-1-3-identity",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
          }
        ],
        "assumptions": []
      },
      {
        "key": "formula",
        "label": "Specified homomorphism",
        "value": "Identify x with X = x⁰I + x¹σ₁ + x²σ₂ + x³σ₃, a Hermitian 2 × 2 matrix. Send X to AXA† for A ∈ SL(2,C). Then det X = (x⁰)² − (x¹)² − (x²)² − (x³)² is preserved.",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
          }
        ],
        "assumptions": []
      },
      {
        "key": "kernel",
        "label": "Kernel",
        "value": "{±I} ≅ C₂",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
          }
        ],
        "assumptions": []
      },
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        "key": "image",
        "label": "Image",
        "value": "SO₀(1,3)",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
          }
        ],
        "assumptions": []
      },
      {
        "key": "injective",
        "label": "Injective",
        "value": "No",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
          }
        ],
        "assumptions": []
      },
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        "key": "surjective",
        "label": "Surjective",
        "value": "Yes",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
          }
        ],
        "assumptions": []
      },
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        "key": "covering-degree",
        "label": "Covering degree",
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        "evidence": [
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            "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
          }
        ],
        "assumptions": []
      }
    ],
    "sections": [
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        "id": "meaning",
        "title": "Global and representation consequences",
        "paragraphs": [
          {
            "text": "The connected source maps onto the identity component of the Lorentz group. It does not cover time reversal or spatial parity components. SL(2,C) is simply connected as a real Lie group.",
            "evidence": [
              {
                "source": "garrett-isogenies",
                "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
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                "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
              }
            ],
            "assumptions": []
          },
          {
            "text": "Preservation of det X follows from det A = 1. If AXA† = X for every Hermitian X, then A is scalar and equals ±I. The derivative is an isomorphism of six-dimensional real Lie algebras; connectedness identifies the image as SO₀(1,3).",
            "evidence": [
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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-global-forms",
                "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)"
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            "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
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      "image": "SO₀(1,3)",
      "injective": false,
      "surjective": true
    }
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      "value": "Identify x with X = x⁰I + x¹σ₁ + x²σ₂ + x³σ₃, a Hermitian 2 × 2 matrix. Send X to AXA† for A ∈ SL(2,C). Then det X = (x⁰)² − (x¹)² − (x²)² − (x³)² is preserved.",
      "evidence": [
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          "source": "garrett-isogenies",
          "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
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          "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
        }
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      "key": "codomain",
      "label": "Codomain",
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    {
      "id": "map:sl-2-c-to-lorentz#formula",
      "subject": "map:sl-2-c-to-lorentz",
      "key": "formula",
      "label": "Specified homomorphism",
      "value": "Identify x with X = x⁰I + x¹σ₁ + x²σ₂ + x³σ₃, a Hermitian 2 × 2 matrix. Send X to AXA† for A ∈ SL(2,C). Then det X = (x⁰)² − (x¹)² − (x²)² − (x³)² is preserved.",
      "evidence": [
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          "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
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      "assumptions": [],
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      "revision": "2026.11-workbench.4"
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      "key": "kernel",
      "label": "Kernel",
      "value": "{±I} ≅ C₂",
      "evidence": [
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          "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
        }
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      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:sl-2-c-to-lorentz#image",
      "subject": "map:sl-2-c-to-lorentz",
      "key": "image",
      "label": "Image",
      "value": "SO₀(1,3)",
      "evidence": [
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          "source": "garrett-isogenies",
          "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
        }
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      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:sl-2-c-to-lorentz#injective",
      "subject": "map:sl-2-c-to-lorentz",
      "key": "injective",
      "label": "Injective",
      "value": "No",
      "evidence": [
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          "source": "garrett-isogenies",
          "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
        }
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      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:sl-2-c-to-lorentz#surjective",
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      "key": "surjective",
      "label": "Surjective",
      "value": "Yes",
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          "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
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      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:sl-2-c-to-lorentz#covering-degree",
      "subject": "map:sl-2-c-to-lorentz",
      "key": "covering-degree",
      "label": "Covering degree",
      "value": "2",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:sl-2-c-to-lorentz#meaning-1",
      "subject": "map:sl-2-c-to-lorentz",
      "key": "meaning-1",
      "label": "Global and representation consequences",
      "value": "The connected source maps onto the identity component of the Lorentz group. It does not cover time reversal or spatial parity components. SL(2,C) is simply connected as a real Lie group.",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
        },
        {
          "source": "etingof-global-forms",
          "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
        }
      ],
      "assumptions": [],
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      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:sl-2-c-to-lorentz#meaning-2",
      "subject": "map:sl-2-c-to-lorentz",
      "key": "meaning-2",
      "label": "Global and representation consequences",
      "value": "Preservation of det X follows from det A = 1. If AXA† = X for every Hermitian X, then A is scalar and equals ±I. The derivative is an isomorphism of six-dimensional real Lie algebras; connectedness identifies the image as SO₀(1,3).",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
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          "source": "etingof-global-forms",
          "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)"
        }
      ],
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        "inputs": [
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      "key": "relation-2",
      "label": "Codomain group",
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          "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)"
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      "relationship": "has-codomain",
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  "sources": [
    {
      "id": "etingof-lie-notes",
      "title": "Lie Groups and Lie Algebras",
      "authors": [
        "Pavel Etingof"
      ],
      "url": "https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec_full.pdf",
      "kind": "lecture-notes",
      "locatorHint": "MIT 18.755, Spring 2024. Cite section numbers and printed pages.",
      "licenseNote": "Linked MIT OpenCourseWare lecture notes. Original summaries only; website content is not offered as a reproduction of these notes."
    },
    {
      "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",
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      "licenseNote": "Linked MIT OpenCourseWare notes; global-form and representation criteria are restated."
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      "id": "garrett-isogenies",
      "title": "Sporadic isogenies to orthogonal groups",
      "authors": [
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      "kind": "lecture-notes",
      "locatorHint": "May 7, 2015 notes, Sections 2.1–2.11. The linked file is an accessible mirror of the author’s notes.",
      "licenseNote": "Author’s lecture notes linked at an accessible mirror. No redistribution license is inferred; matrix constructions are explained independently."
    }
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