{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:9bd75d12-0eb0-4d97-ab55-23361589bdab", "labels": { "zh-CN": "群作用", "en": "group action" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "6572899b926fd34a211648e4a76d7186e67877b3f3d54bbcffa774497b93d195", "occurrences": [ { "concept_id": "urn:uuid:9bd75d12-0eb0-4d97-ab55-23361589bdab", "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-2114245f91c19109af2f-zh", "chapter": "7", "section": "7.4", "offset": 10034, "range": [ 18114, 18410 ], "source_revision": "dece3546a425ea2c52b709472898bb8128e5d5d751614a636ab79e1e1bbfe61c", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "群作用(group action)", "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-d2dcdc9bfbf6a1a6f35d2c6a" }, { "concept_id": "urn:uuid:9bd75d12-0eb0-4d97-ab55-23361589bdab", "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-2114245f91c19109af2f-en", "chapter": "7", "section": "7.4", "offset": 12131, "range": [ 19544, 19835 ], "source_revision": "56ee7f1727deaa19e151a716a8ad06b4eb7d62437a0e4e97e582b9d242bc09a3", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "group action", "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-a6f67188aefb281014d4e4c6" }, { "concept_id": "urn:uuid:9bd75d12-0eb0-4d97-ab55-23361589bdab", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-16/01-the-symplectic-group", "anchor": "equation-16-9", "chapter": "16", "section": "16.1", "offset": 6725, "range": [ 14492, 14753 ], "source_revision": "5d0feb2b6d5a17ec427c9df5dc00639e77e4f399daa61206841492326af9d8d6", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "群作用(group action)", "context": { "range": [ 14383, 14866 ], "quote": "其中（对于写成列向量的 $m \\in M$）$e ^ { - t L } \\cdot m$ 是乘以矩阵 $e ^ { - t L }$。在写成列向量的线性函数 $l \\in \\mathcal { M }$ 上，同一个[群作用(group action)](../../../../../concepts/group-theory/group-action)将 $l$ 变为 $e ^ { t L } l$，并对 $\\mathcal { M }$ 的基向量 $q , p$ 作用为", "quote_sha256": "64b988e4d8b6e743be16c71e92070466f8a1d04ecf5e4e26ea0ed3f077d0e4e5", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-16/01-the-symplectic-group", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-87b6b8e4d8e4773ed7bdb65e" }, { "concept_id": "urn:uuid:9bd75d12-0eb0-4d97-ab55-23361589bdab", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-16/01-the-symplectic-group", "anchor": "equation-16-9", "chapter": "16", "section": "16.1", "offset": 8526, "range": [ 15459, 15715 ], "source_revision": "ed2414a74ed2fd73363cf684a60f2b8e6245a4dc84ed67db852efd43fec8ffec", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "group action", "context": { "range": [ 15257, 15851 ], "quote": "where (for $m \\in M$ written as column vectors) $e ^ { - t L } \\cdot m$ is multiplication by the matrix $e ^ { - t L }$ . On linear functions $l \\in \\mathcal { M }$ written as column vectors, the same [group action](../../../../../concepts/group-theory/group-action) takes l to $e ^ { t L } l$ and acts on basis vectors $q , p$ of $\\mathcal { M }$ by", "quote_sha256": "fc4852f355bf7473ca13afdf55f1859164472a0105312c0c078ec8c693ae28cb", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-16/01-the-symplectic-group", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-1934cc8a4a99064176c27676" } ], "registered_positions": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space", "lang": "en", "section": "7.4", "signal": "term-mark", "label": "group action", "english": "group action", "offset": 8938 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "lang": "zh-CN", "section": "7.4", "signal": "term-mark", "label": "群作用", "english": "group action", "offset": 6876 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-16/01-the-symplectic-group", "lang": "en", "section": "16.1", "signal": "term-mark", "label": "group action", "english": "group action", "offset": 7832 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-16/01-the-symplectic-group", "lang": "zh-CN", "section": "16.1", "signal": "term-mark", "label": "群作用", "english": "group action", "offset": 6030 } ], "statements": [], "resolved_references": [ { "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-d2dcdc9bfbf6a1a6f35d2c6a" }, { "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-a6f67188aefb281014d4e4c6" } ], "reading_order": [ "occ-d2dcdc9bfbf6a1a6f35d2c6a", "occ-a6f67188aefb281014d4e4c6", "occ-87b6b8e4d8e4773ed7bdb65e", "occ-1934cc8a4a99064176c27676" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 4, "located_occurrences": 4, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "d4cf13a1ed6518c663a46346de3a88a0baa96521cafa1d59958ef87e014c59bd", "optional_enrichments": [] }