{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:e975605a-7577-4615-b024-f78011266eb4", "labels": { "zh-CN": "四元数", "en": "quaternion" }, "alternative_labels": [ { "lang": "en", "value": "quaternions" } ], "scope_note": "", "identity_status": "active", "identity_revision": "97311e01742826eaea528da836f3be6dd23debfd7a02a502bf6a5202378cefdc", "occurrences": [ { "concept_id": "urn:uuid:e975605a-7577-4615-b024-f78011266eb4", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-06/03-quaternions-and-clifford-algebras", "anchor": "concept-6-quaternion", "chapter": "6", "section": "6.3", "offset": 893, "range": [ 5511, 5736 ], "source_revision": "576f66d635cff4aa7f260396a55ab322c79487e370dd4763ae8af64c8d308bdb", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "四元数(quaternion)", "context": { "range": [ 5488, 5775 ], "quote": "其余恒等式可以类似方式推出。这个代数的元素称为四元数(quaternion)；它们构成一个非交换代数，因为$i j - j i = 2 k \\neq 0$", "quote_sha256": "2762c52207353bca233a83bcecc1a6cbd66261d11c205352fd9fe66b56d67078", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-06/03-quaternions-and-clifford-algebras", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-a0dfd20f21791c7652db7cad" }, { "concept_id": "urn:uuid:e975605a-7577-4615-b024-f78011266eb4", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-06/03-quaternions-and-clifford-algebras", "anchor": "first-972a5aa65f6fb37f163f-en", "chapter": "6", "section": "6.3", "offset": 1261, "range": [ 5367, 5597 ], "source_revision": "cdcd79c511136b0ea534aa44839633fd1b78a9b0561c11f8e5a1c2073b5bfc9c", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "quaternions", "context": { "range": [ 5281, 5665 ], "quote": "The other identities follow in a similar way. The elements of this algebra are called quaternions; they form a non-commutative algebra since $i j - j i = 2 k \\neq 0$", "quote_sha256": "f868994d1f11ca6983e299a95e4441088b2f99730dab82378afc052eae3a44a5", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-06/03-quaternions-and-clifford-algebras", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-e39e8c4214ade0c0244c5bad" }, { "concept_id": "urn:uuid:e975605a-7577-4615-b024-f78011266eb4", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions", "anchor": "concept-6-quaternion", "chapter": "6", "section": "6.2", "offset": 3426, "range": [ 11717, 12034 ], "source_revision": "d5776d5bdae66f74cdbd2aaeda0ef4ffb4148f08eeb21a72c16dd61bab582129", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "四元数(quaternion)", "context": { "range": [ 11627, 12246 ], "quote": "给定 $\\operatorname{Spin}(n)$ 的构造，还需显式构造同态 $\\Phi$，并证明其导数 $\\Phi\\prime$ 是李代数同构。这里最简单的构造使用单位长度[四元数(quaternion)](../../../../../concepts/algebra/quaternion)群 $\\mathrm{Sp}(1)$：$\\operatorname{Spin}(3)=\\mathrm{Sp}(1)$，而 $\\operatorname{Spin}(4)=\\mathrm{Sp}(1)\\times\\mathrm{Sp}(1)$。将四元数与复数对等同，可得 $\\mathrm{Sp}(1)=\\mathrm{SU}(2)$。", "quote_sha256": "0f1c83d3c1204def6ea922544e9569941e1152e80f57da4905385b64904e7578", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-846db1a07c0db171869cc93f" }, { "concept_id": "urn:uuid:e975605a-7577-4615-b024-f78011266eb4", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions", "anchor": "source-6-quaternion", "chapter": "6", "section": "6.2", "offset": 4089, "range": [ 12380, 12632 ], "source_revision": "d5776d5bdae66f74cdbd2aaeda0ef4ffb4148f08eeb21a72c16dd61bab582129", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "四元数(quaternion)", "context": { "range": [ 12379, 12732 ], "quote": "[四元数(quaternion)](../../../../../concepts/algebra/quaternion)（记作 $\\mathbb H$）构成一个数系，是复数系的推广。其元素 $q\\in\\mathbb H$ 可写为：", "quote_sha256": "e995e497d413e3c6db20dfecf9275e116f91f2a4ea6f3040306aec6831a629a8", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-e67d15546b6c4acacd680cd9" }, { "concept_id": "urn:uuid:e975605a-7577-4615-b024-f78011266eb4", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions", "anchor": "concept-6-quaternion", "chapter": "6", "section": "6.2", "offset": 4544, "range": [ 11993, 12306 ], "source_revision": "525b7bc4b6973b05449bbbbcbdef1c067e7411218e20b20cb388b95687d1c4d8", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "quaternions", "context": { "range": [ 11728, 12555 ], "quote": "Given this construction of $\\operatorname{Spin}(n)$, we also need to construct the homomorphism $\\Phi$ explicitly and show that its derivative $\\Phi\\prime$ is an isomorphism of Lie algebras. The simplest construction uses the group $\\mathrm{Sp}(1)$ of unit-length [quaternions](../../../../../concepts/algebra/quaternion): $\\operatorname{Spin}(3)=\\mathrm{Sp}(1)$ and $\\operatorname{Spin}(4)=\\mathrm{Sp}(1)\\times\\mathrm{Sp}(1)$. Identifying quaternions with pairs of complex numbers shows that $\\mathrm{Sp}(1)=\\mathrm{SU}(2)$.", "quote_sha256": "dcc13b2951e6d797bc0e19cdffa7046473b1e5fa8691b06f631812eb58ff2612", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-77f01602ec81e13758fee161" }, { "concept_id": "urn:uuid:e975605a-7577-4615-b024-f78011266eb4", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions", "anchor": "source-6-quaternion", "chapter": "6", "section": "6.2", "offset": 5252, "range": [ 12701, 12949 ], "source_revision": "525b7bc4b6973b05449bbbbcbdef1c067e7411218e20b20cb388b95687d1c4d8", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "quaternions", "context": { "range": [ 12696, 13126 ], "quote": "The [quaternions](../../../../../concepts/algebra/quaternion) form a number system, denoted by $\\mathbb H$, that generalizes the complex numbers. Its elements $q\\in\\mathbb H$ can be written as:", "quote_sha256": "3c58d3033aaff0fad652c18eec0cadf2dc5ea088925535a675d1430cdd265c52", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-dacbe8db37642e4c2fc6ce1f" }, { "concept_id": "urn:uuid:e975605a-7577-4615-b024-f78011266eb4", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-28/05-real-cliford-algebras", "anchor": "source-28-2-real-clifford-algebra-arbitrary-signature", "chapter": "28", "section": "28.2", "offset": 2618, "range": [ 9403, 9655 ], "source_revision": "0dc0c1bd23025c774c4a68979991f28351ea92df3276584446b85a848929b601", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "四元数(quaternion)", "context": { "range": [ 9391, 9723 ], "quote": " 这个四维实代数可与[四元数(quaternion)](../../../../../concepts/algebra/quaternion)代数 $\\mathbb H$ 等同，对应关系为", "quote_sha256": "3edb5b2ca1146d07826ef755d5868a41f18cb6c5980927b4804a4d82076875c3", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-28/05-real-cliford-algebras", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-7edab64add570c5153e4f21e" } ], "registered_positions": [ { "route": "en/chapter-06/03-quaternions-and-clifford-algebras", "lang": "en", "section": "6.3", "signal": "term-mark", "label": "quaternions", "english": "quaternions", "offset": 1261 }, { "route": "zh/chapter-06/03-quaternions-and-clifford-algebras", "lang": "zh-CN", "section": "6.3", "signal": "term-mark", "label": "四元数", "english": "quaternion", "offset": 893 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions", "lang": "en", "section": "6.2", "signal": "term-mark", "label": "quaternions", "english": "quaternions", "offset": 3452 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions", "lang": "en", "section": "6.2", "signal": "term-mark", "label": "quaternions", "english": "quaternions", "offset": 4004 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions", "lang": "zh-CN", "section": "6.2", "signal": "term-mark", "label": "四元数", "english": "quaternion", "offset": 2490 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions", "lang": "zh-CN", "section": "6.2", "signal": "term-mark", "label": "四元数", "english": "quaternion", "offset": 2997 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-28/05-real-cliford-algebras", "lang": "zh-CN", "section": "28.2", "signal": "term-mark", "label": "四元数", "english": "quaternion", "offset": 2080 } ], "statements": [], "resolved_references": [], "reading_order": [ "occ-a0dfd20f21791c7652db7cad", "occ-e39e8c4214ade0c0244c5bad", "occ-846db1a07c0db171869cc93f", "occ-e67d15546b6c4acacd680cd9", "occ-77f01602ec81e13758fee161", "occ-dacbe8db37642e4c2fc6ce1f", "occ-7edab64add570c5153e4f21e" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 7, "located_occurrences": 7, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "7453de4fc971dd728527afd9aeb736dda5b56f0a3443352739b1466fef3d11ea", "optional_enrichments": [] }