{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:6328536c-e4a4-4277-b275-0a8a7146d152", "labels": { "zh-CN": "莫比乌斯变换", "en": "Möbius transformation" }, "alternative_labels": [ { "lang": "en", "value": "Möbius transformations" } ], "scope_note": "", "identity_status": "active", "identity_revision": "0bbb5d90608f4aabe694be2c6571001b5cfa3eefb9645085aeb2930633b29104", "occurrences": [ { "concept_id": "urn:uuid:6328536c-e4a4-4277-b275-0a8a7146d152", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-02/05-normal-subgroups-and-factor-groups", "anchor": "concept-2-mobius-transformation", "chapter": "2", "section": "2.5", "offset": 10765, "range": [ 15487, 15737 ], "source_revision": "fee321b0500c365016e4a1e97ab854a0117b68f310294f13cc853414839be4d8", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "莫比乌斯变换(Möbius transformation)", "context": { "range": [ 15407, 16366 ], "quote": "> [!exercise] 习题 2.16 \n>\n> 莫比乌斯变换(Möbius transformation)是复映射\n>\n> $$\n> z\\mapsto z'=\\frac{az+b}{cz+d},\\qquad a,b,c,d\\in\\mathbb{C},\\quad ad-bc=1.\n> $$\n>\n> (a) 证明这些映射是扩充复平面（包含点 $z=\\infty$）上的一一到上变换，并写出分别由常数 $(a,b,c,d)$ 和 $(a',b',c',d')$ 确定的任意两个变换的复合。\n>\n> (b) 证明它们组成一个群，称为莫比乌斯群(Möbius group)。\n>\n> (c) 证明把幺模矩阵 $\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}$ 映到上述莫比乌斯变换的映射 $\\mu$，是从 $SL(2,\\mathbb{C})$ 到莫比乌斯群的同态，其核为 $\\{\\mathsf{I},-\\mathsf{I}\\}$。也就是说，莫比乌斯群同构于 $SL(2,\\mathbb{C})/\\mathbb{Z}_2$。", "quote_sha256": "220fccd581f6db69a20e0b9bac480dfb93e0d0991a5093d760a65601d230b5df", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-02/05-normal-subgroups-and-factor-groups", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-ef980f2360f0587ad7106084" }, { "concept_id": "urn:uuid:6328536c-e4a4-4277-b275-0a8a7146d152", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "anchor": "first-5ac38295dc438d0557a8-zh", "chapter": "7", "section": "7.4", "offset": 9609, "range": [ 17689, 18002 ], "source_revision": "dece3546a425ea2c52b709472898bb8128e5d5d751614a636ab79e1e1bbfe61c", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "莫比乌斯变换(Möbius transformation)", "context": { "range": [ 17663, 18689 ], "quote": "若矩阵行列式非零，相应变换便可逆。这些保角变换称为[莫比乌斯变换(Möbius transformation)](../../../../../concepts/geometry/mobius-transformation.md)。在[第 40 章](../chapter-40/index#section-40)中，我们会看到这种[群作用(group action)](../../../../../concepts/group-theory/group-action.md)出现在狭义相对论中；它对球面的作用可以理解为对光线空间的变换。当矩阵属于 $SU(2)$ 时，其元素满足 $\\gamma=-\\overline\\beta$、$\\delta=\\overline\\alpha$ 以及 $|\\alpha|^2+|\\beta|^2=1$。此时球面上的对应变换就是 $\\mathbf R^3$ 中球面的旋转。这也从另一角度说明 $SU(2)=\\operatorname{Spin}(3)$ 是旋转群 $SO(3)$ 的双覆盖。", "quote_sha256": "5ee254b687997d049d081912719f120c764f3082f952c4ebba186a7dbb1bfc65", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-5a871c41fb595853d7c1d67d" }, { "concept_id": "urn:uuid:6328536c-e4a4-4277-b275-0a8a7146d152", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space", "anchor": "first-e2a6486a1efc2e914542-en", "chapter": "7", "section": "7.4", "offset": 11696, "range": [ 19109, 19415 ], "source_revision": "56ee7f1727deaa19e151a716a8ad06b4eb7d62437a0e4e97e582b9d242bc09a3", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "Möbius transformations", "context": { "range": [ 18952, 20303 ], "quote": "If the matrix has nonzero determinant, the transformation is invertible. These are conformal (angle-preserving) transformations of the complex plane called [Möbius transformations](../../../../../concepts/geometry/mobius-transformation.md). In [Chapter 40](../chapter-40/index#section-40), we will see this [group action](../../../../../concepts/group-theory/group-action.md) in special relativity, where its action on the sphere can be understood as a transformation of the space of light rays. For a matrix in $SU(2)$, the entries satisfy $\\gamma=-\\overline\\beta$, $\\delta=\\overline\\alpha$, and $|\\alpha|^2+|\\beta|^2=1$. The corresponding transformation rotates the sphere in $\\mathbf R^3$, giving another way to understand $SU(2)=\\operatorname{Spin}(3)$ as the double cover of $SO(3)$.", "quote_sha256": "027d7af7ece7a58f313ffcb85bc6065b66f86d60479913efc43c403c3bc744c0", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-98e2e67d44a94923da98448b" } ], "registered_positions": [ { "route": "zh/chapter-02/05-normal-subgroups-and-factor-groups", "lang": "zh-CN", "section": "2.5", "signal": "term-mark", "label": "莫比乌斯变换", "english": "Möbius transformation", "offset": 9460 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space", "lang": "en", "section": "7.4", "signal": "term-mark", "label": "Möbius transformations", "english": "Möbius transformations", "offset": 8694 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "lang": "zh-CN", "section": "7.4", "signal": "term-mark", "label": "莫比乌斯变换", "english": "Möbius transformation", "offset": 6642 } ], "statements": [], "resolved_references": [ { "raw": "href=\"#problem-2-16\"", "href": "#problem-2-16", "range": [ 15449, 15469 ], "relation": "references", "target_route": "zh/chapter-02/05-normal-subgroups-and-factor-groups", "target_anchor": "problem-2-16", "resolved": true, "reason": null, "source_revision": "fee321b0500c365016e4a1e97ab854a0117b68f310294f13cc853414839be4d8", "source_route": "zh/chapter-02/05-normal-subgroups-and-factor-groups", "occurrence_id": "occ-ef980f2360f0587ad7106084" }, { "raw": "[第 40 章](../chapter-40/index#section-40)", "href": "../chapter-40/index#section-40", "range": [ 18064, 18104 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/zh/chapter-40/index", "target_anchor": "section-40", "resolved": true, "reason": null, "source_revision": "dece3546a425ea2c52b709472898bb8128e5d5d751614a636ab79e1e1bbfe61c", "source_route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "occurrence_id": "occ-5a871c41fb595853d7c1d67d" }, { "raw": "[Chapter 40](../chapter-40/index#section-40)", "href": "../chapter-40/index#section-40", "range": [ 19480, 19524 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/en/chapter-40/index", "target_anchor": "section-40", "resolved": true, "reason": null, "source_revision": "56ee7f1727deaa19e151a716a8ad06b4eb7d62437a0e4e97e582b9d242bc09a3", "source_route": "books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space", "occurrence_id": "occ-98e2e67d44a94923da98448b" } ], "reading_order": [ "occ-ef980f2360f0587ad7106084", "occ-5a871c41fb595853d7c1d67d", "occ-98e2e67d44a94923da98448b" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 3, "located_occurrences": 3, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "637c5330dfb7e8386c7dd76cbaca920930d3a294732d427c2478c2146ffed0a1", "optional_enrichments": [] }