{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:87107086-71e4-4b8b-899e-10c24486072b", "labels": { "zh-CN": "雅可比恒等式", "en": "Jacobi identity" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "0f8ab4f0941edc242412639d31550fc39464affeb09641d393f8ad2b0ae2fa67", "occurrences": [ { "concept_id": "urn:uuid:87107086-71e4-4b8b-899e-10c24486072b", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-05/04-lie-algebra-representations", "anchor": "first-5a798c9794bd7dbacc1a-zh", "chapter": "5", "section": "5.4", "offset": 16087, "range": [ 26099, 26402 ], "source_revision": "7b44f63f358be0b87925985afa44ebc92b5cb64195e88543ca8324274528e188", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "雅可比恒等式(Jacobi identity)", "context": { "range": [ 26095, 26551 ], "quote": "这称为[雅可比恒等式(Jacobi identity)](../../../../../concepts/lie-theory/jacobi-identity)。直接展开矩阵乘积也能更简单地推出这一恒等式；不过这里还可以从一个更抽象的角度理解它为何成立：伴随表示的存在。利用对易子的反对称性重新排列各项，还能得到其他等价形式；例如，对循环置换求和可得：", "quote_sha256": "fedf6d7fb8cc559249e8bd8b03135ce0f7aa49e8765ee1203bb84dbc9bbb24dc", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-05/04-lie-algebra-representations", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-667a4b2afc31b3e0f5dfa743" }, { "concept_id": "urn:uuid:87107086-71e4-4b8b-899e-10c24486072b", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-05/04-lie-algebra-representations", "anchor": "first-5a798c9794bd7dbacc1a-en", "chapter": "5", "section": "5.4", "offset": 17535, "range": [ 26771, 27066 ], "source_revision": "61c42cc7f275c26a6874d48870eeec0207a69553f4f81e2310258160b4921899", "source_category": "original_transcription", "review_status": "ai-transcription-compared", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "Jacobi identity", "context": { "range": [ 26751, 27423 ], "quote": "This is called the [Jacobi identity](../../../../../concepts/lie-theory/jacobi-identity). It could be derived more simply by expanding the matrix products, but here we see a more abstract reason it holds: the existence of the adjoint representation. Using antisymmetry of the commutator to rearrange the terms gives other equivalent forms; for example, summing over cyclic permutations yields", "quote_sha256": "d73e1a7c7d6c105dafdf4534f64361648420dc2e116261ec34b47512580cc5e3", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-05/04-lie-algebra-representations", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-7f177abc1d9a576cc32de094" }, { "concept_id": "urn:uuid:87107086-71e4-4b8b-899e-10c24486072b", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-05/04-lie-algebra-representations", "anchor": "source-block-5191", "chapter": "5", "section": "5.4", "offset": 17704, "range": [ 27716, 27984 ], "source_revision": "7b44f63f358be0b87925985afa44ebc92b5cb64195e88543ca8324274528e188", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "雅可比恒等式(Jacobi identity)", "context": { "range": [ 26740, 28159 ], "quote": "\n\n> [!definition] 定义 · 抽象李代数\n> \n>\n> 定义（抽象李代数）。域 $k$ 上的一个 抽象李代数(abstract Lie algebra) 是 $k$ 上的一个向量空间 $A$，带有一个双线性运算\n>\n> \n>\n> $$\n> [\\cdot,\\cdot]:(X,Y)\\in A\\times A\\mapsto[X,Y]\\in A\n> $$\n>\n> \n>\n> 满足\n>\n> 1. 反对称性：\n>\n> \n>\n> $$\n> [X,Y]=-[Y,X]\n> $$\n>\n> 2. [雅可比恒等式(Jacobi identity)](../../../../../concepts/lie-theory/jacobi-identity)：\n>\n> \n>\n> $$\n> [[X,Y],Z]+[[Z,X],Y]+[[Y,Z],X]=0\n> $$\n", "quote_sha256": "9869d7bc3d8e9b36b8d240e11f61d867e71fe126eea4ff2b5fa6a51e83c66a84", "environment": "definition", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-05/04-lie-algebra-representations", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-2b84121c75b49a9b6a0197c8" } ], "registered_positions": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-05/04-lie-algebra-representations", "lang": "en", "section": "5.4", "signal": "term-mark", "label": "Jacobi identity", "english": "Jacobi identity", "offset": 14306 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-05/04-lie-algebra-representations", "lang": "zh-CN", "section": "5.4", "signal": "term-mark", "label": "雅可比恒等式", "english": "Jacobi identity", "offset": 12541 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-05/04-lie-algebra-representations", "lang": "zh-CN", "section": "5.4", "signal": "term-mark", "label": "雅可比恒等式", "english": "Jacobi identity", "offset": 13740 } ], "statements": [], "resolved_references": [], "reading_order": [ "occ-667a4b2afc31b3e0f5dfa743", "occ-7f177abc1d9a576cc32de094", "occ-2b84121c75b49a9b6a0197c8" ], "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": "e7144a0a631ae5e019487ae451a3900d4892a3c10f0c57364607105d49026628", "optional_enrichments": [] }