{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:1afbc47e-4026-42fb-a8c1-d2b6a02638c4", "labels": { "zh-CN": "单位四元数", "en": "unit quaternion" }, "alternative_labels": [ { "lang": "en", "value": "unit quaternions" } ], "scope_note": "", "identity_status": "active", "identity_revision": "8d62c44b63107b23994dfcb4504315f7458baddf655c2347229a3ef0c126252a", "occurrences": [ { "concept_id": "urn:uuid:1afbc47e-4026-42fb-a8c1-d2b6a02638c4", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/07-a-summary", "anchor": "concept-6-3-quaternion", "chapter": "6", "section": "6.3", "offset": 2595, "range": [ 11938, 12267 ], "source_revision": "51224e8d409a684fb67417edf938c1deffb0aa708773316223994618d7a07300", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "单位四元数(unit quaternions)", "context": { "range": [ 10266, 12817 ], "quote": null, "quote_sha256": "ccc6c8c7f3b76989ce54f30bbc227a5304b6c0410f33b5205f652eb820bc410b", "environment": null, "excerpt_omitted": true }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-06/07-a-summary", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-77b8947b40967383e786ae9a" }, { "concept_id": "urn:uuid:1afbc47e-4026-42fb-a8c1-d2b6a02638c4", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-06/07-a-summary", "anchor": "concept-6-3-quaternion", "chapter": "6", "section": "6.3", "offset": 2745, "range": [ 11254, 11576 ], "source_revision": "c6959dd1685931e5d5c3b3ad92a453f6e8e23f74fb6df74bd7839b5bfd7d43ac", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "unit quaternions", "context": { "range": [ 9500, 12157 ], "quote": null, "quote_sha256": "51760142b0b6b47418ad93766ff4c4d84f786e42d2663d2b951cfe7cd1240ff2", "environment": null, "excerpt_omitted": true }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/07-a-summary", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-e6df2e19280a8a85b917820d" }, { "concept_id": "urn:uuid:1afbc47e-4026-42fb-a8c1-d2b6a02638c4", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/01-the-spinor-representation", "anchor": "concept-7-1-quaternion", "chapter": "7", "section": "7.1", "offset": 2389, "range": [ 10547, 10875 ], "source_revision": "3bf7a61152ec8da38212ac86935b672a5739debafd35daed43c2f6c6fd6e56ef", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "单位四元数(unit quaternion)", "context": { "range": [ 10389, 11072 ], "quote": "在 $\\mathrm{SU}(2)$ 和 $\\mathrm{Sp}(1)$ 的情形下，还存在比伴随表示更简单的非平凡不可约表示：一是 $\\mathrm{SU}(2)$ 中的 $2\\times2$ 复矩阵按矩阵乘法作用于列向量 $\\mathbf{C}^2$ 所给出的表示；二是 $\\mathrm{Sp}(1)$ 中的[单位四元数(unit quaternion)](../../../../../concepts/algebra/quaternion)按标量乘法作用于 $\\mathbf{H}$ 所给出的表示。选定 $\\mathbf{C}^2$ 与 $\\mathbf{H}$ 之间的识别后，这两个表示便是同构群在 $\\mathbf{C}^2$ 上的同构表示。为便于计算，我们将使用 $\\mathrm{SU}(2)$ 及其复矩阵，而不直接处理四元数。于是有：", "quote_sha256": "4a0fdd434a55d9ce24a7aa6365d7a42210ae11d88b965e5ef0325d64711fbba8", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-07/01-the-spinor-representation", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-b3d5d8f95859f011d69b30b6" }, { "concept_id": "urn:uuid:1afbc47e-4026-42fb-a8c1-d2b6a02638c4", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-07/01-the-spinor-representation", "anchor": "concept-7-1-quaternion", "chapter": "7", "section": "7.1", "offset": 2643, "range": [ 9965, 10287 ], "source_revision": "15383506dc3beaec4ce03121843583ac8e897a21ef7acf46a88bffa7c86291e9", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "unit quaternions", "context": { "range": [ 9668, 10666 ], "quote": "In the $\\mathrm{SU}(2)$ and $\\mathrm{Sp}(1)$ cases, there is an even simpler nontrivial irreducible representation than the adjoint. One is given by the $2\\times2$ complex matrices in $\\mathrm{SU}(2)$ acting by matrix multiplication on column vectors in $\\mathbf{C}^2$; the other is given by the [unit quaternions](../../../../../concepts/algebra/quaternion) in $\\mathrm{Sp}(1)$ acting by scalar multiplication on $\\mathbf{H}$. After identifying $\\mathbf{C}^2$ with $\\mathbf{H}$, these are isomorphic representations on $\\mathbf{C}^2$ of isomorphic groups. For convenience in calculations, we will use $\\mathrm{SU}(2)$ and its complex matrices rather than deal with quaternions. 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