{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:b4e48583-4b04-48fc-83d8-e77fc248bcb3", "labels": { "zh-CN": "斜伴随", "en": "skew-adjoint" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "7099649d79516d17229cbee64ef17ccdbdafa5137dbf4ab4a601f38fac525806", "occurrences": [ { "concept_id": "urn:uuid:b4e48583-4b04-48fc-83d8-e77fc248bcb3", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-01/04-representations-and-quantum-mechanics", "anchor": "concept-1-skew-adjoint", "chapter": "1", "section": "1.4", "offset": 1275, "range": [ 9490, 9791 ], "source_revision": "651b93af49354643df29235b5d5a8624db703c2491d04f67cb8fa292c50899ca", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "斜伴随(skew-adjoint)", "context": { "range": [ 9382, 9794 ], "quote": "其中 $A$ 也是一个接近零矩阵的矩阵。我们将更详细地研究这种情形，并通过大量例子说明：如果 $\\pi(g)$ 是酉矩阵（即属于子群 $U(n)\\subset GL(n,\\mathbf{C})$），那么 $A$ 就是斜伴随(skew-adjoint)矩阵：", "quote_sha256": "26f1e5a5c07d319462b1189e9f9954809b1765c240819d4e71f6b89763684c17", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-01/04-representations-and-quantum-mechanics", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-05e460d3b165b2969e13aeb6" }, { "concept_id": "urn:uuid:b4e48583-4b04-48fc-83d8-e77fc248bcb3", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-01/04-representations-and-quantum-mechanics", "anchor": "concept-1-skew-adjoint", "chapter": "1", "section": "1.4", "offset": 1674, "range": [ 8438, 8734 ], "source_revision": "dda88312d4b3b82575527c09653b3c1ef9836bc6fbb287f6909d0a6223a09af1", "source_category": "original_transcription", "review_status": "ai-transcription-compared", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "skew-adjoint", "context": { "range": [ 8158, 8735 ], "quote": "where $A$ is also a matrix, close to the zero matrix. We will study this situation in much more detail and work extensively with examples, showing in particular that if $\\pi ( g )$ is unitary (i.e. , in the subgroup $U ( n ) \\subset GL( n , \\mathbf { C } ) )$ ), then $A$ will be skew-adjoint:", "quote_sha256": "a7b16be59010f8a6e15cc5906e358aaa92a99124529342c84f31420f8d834415", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-01/04-representations-and-quantum-mechanics", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-746c74e64b7b11a220883b93" }, { "concept_id": "urn:uuid:b4e48583-4b04-48fc-83d8-e77fc248bcb3", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-04/05-adjoint-operators", "anchor": "concept-4-skew-adjoint-operator", "chapter": "4", "section": "4.5", "offset": 2513, "range": [ 7969, 8246 ], "source_revision": "cd9f99f91376f3deea8f9551c71a0c0de1b0b8de1d032daaa7bb5f1c3a28b7bf", "source_category": "original_transcription", "review_status": "ai-transcription-compared", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "skew-adjoint", "context": { "range": [ 7459, 8275 ], "quote": "In the real case, the matrix of the adjoint is the transpose. A linear transformation $L$ is [self-adjoint] if $L ^ { \\dagger } = L$, and [skew-adjoint] if $L ^ { \\dagger } = - L$.", "quote_sha256": "8d16348327a8a2d381d31581c76f4ec0fb070a18ab90b6f0be7853a44bfbc121", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-04/05-adjoint-operators", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-08a899f8467e000a89a13bb5" }, { "concept_id": "urn:uuid:b4e48583-4b04-48fc-83d8-e77fc248bcb3", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups", "anchor": "source-block-5076", "chapter": "5", "section": "5.2", "offset": 3122, "range": [ 11299, 11564 ], "source_revision": "d27a68e5ce7f74c89a30270d7d93a2543afc658563f090d2f75e8b0d23e73f84", "source_category": "original_transcription", "review_status": "ai-transcription-compared", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "skew-adjoint", "context": { "range": [ 11254, 11679 ], "quote": "so the matrices to be exponentiated must be [skew-adjoint](../../../../../concepts/linear-algebra/skew-adjoint-matrix) (this can also be shown to be sufficient), satisfying", "quote_sha256": "959fb4fbbd3e5368ba0f97c20cf5902f63adceed8e458d0c2ac0a6899df4db3a", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-235c7774c3514e115ca420f2" }, { "concept_id": "urn:uuid:b4e48583-4b04-48fc-83d8-e77fc248bcb3", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups", "anchor": "source-block-5093", "chapter": "5", "section": "5.2", "offset": 8907, "range": [ 17084, 17349 ], "source_revision": "d27a68e5ce7f74c89a30270d7d93a2543afc658563f090d2f75e8b0d23e73f84", "source_category": "original_transcription", "review_status": "ai-transcription-compared", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "skew-adjoint", "context": { "range": [ 16694, 17801 ], "quote": "For the case of the [unitary group](../../../../../concepts/linear-algebra/unitary-group) $U(n)$, the unitarity condition implies that $X$ is [skew-adjoint](../../../../../concepts/linear-algebra/skew-adjoint-matrix) (also called [skew-Hermitian](../../../../../concepts/linear-algebra/skew-adjoint-matrix)), satisfying", "quote_sha256": "f20d997542bec9a0a870c35b07f469f91f16453dc7708bfc430e577516f645aa", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-05938bb0f184ecc0b10a6250" }, { "concept_id": "urn:uuid:b4e48583-4b04-48fc-83d8-e77fc248bcb3", "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": "source-block-5134", "chapter": "5", "section": "5.4", "offset": 4339, "range": [ 13575, 13840 ], "source_revision": "61c42cc7f275c26a6874d48870eeec0207a69553f4f81e2310258160b4921899", "source_category": "original_transcription", "review_status": "ai-transcription-compared", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "skew-adjoint", "context": { "range": [ 13522, 13902 ], "quote": "The representation is unitary when its matrices are [skew-adjoint](../../../../../concepts/linear-algebra/skew-adjoint-matrix).", "quote_sha256": "0fdc5ce8358558d0ed08b0c8e24fa69729abb4a7b2d793e45c064a1514d84e8a", "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-38d77b83d99543fe89e4a3da" } ], "registered_positions": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-01/04-representations-and-quantum-mechanics", "lang": "en", "section": "1.4", "signal": "term-mark", "label": "skew-adjoint", "english": "skew-adjoint", "offset": 1674 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-01/04-representations-and-quantum-mechanics", "lang": "zh-CN", "section": "1.4", "signal": "term-mark", "label": "斜伴随", "english": "skew-adjoint", "offset": 1119 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-04/05-adjoint-operators", "lang": "en", "section": "4.5", "signal": "term-mark", "label": "skew-adjoint", "english": "skew-adjoint", "offset": 2024 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups", "lang": "en", "section": "5.2", "signal": "term-mark", "label": "skew-adjoint", "english": "skew-adjoint", "offset": 2810 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups", "lang": "en", "section": "5.2", "signal": "term-mark", "label": "skew-adjoint", "english": "skew-adjoint", "offset": 7398 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-05/04-lie-algebra-representations", "lang": "en", "section": "5.4", "signal": "term-mark", "label": "skew-adjoint", "english": "skew-adjoint", "offset": 3263 } ], "statements": [], "resolved_references": [], "reading_order": [ "occ-05e460d3b165b2969e13aeb6", "occ-746c74e64b7b11a220883b93", "occ-08a899f8467e000a89a13bb5", "occ-235c7774c3514e115ca420f2", "occ-05938bb0f184ecc0b10a6250", "occ-38d77b83d99543fe89e4a3da" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 6, "located_occurrences": 6, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "6fe23cf63cf96e9b21bc754305b930b68e508c07a39074a3397c8df650d97daf", "optional_enrichments": [] }