{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:475077c6-8a0d-4954-9883-ba29330925d3", "labels": { "zh-CN": "紧辛群（四元数酉群）", "en": "compact symplectic group" }, "alternative_labels": [ { "lang": "en", "value": "symplectic groups" }, { "lang": "zh-CN", "value": "辛群" } ], "scope_note": "§6.2 uses Sp(n) preserving the quaternionic norm; distinct from Sp(2n,R).", "identity_status": "active", "identity_revision": "955d93678e6daeb36d0fc850bea6a58625007411236c9fa3888ddc1c283d76f6", "occurrences": [ { "concept_id": "urn:uuid:475077c6-8a0d-4954-9883-ba29330925d3", "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": "first-bc61ca187a358bb1fe74-zh", "chapter": "6", "section": "6.2", "offset": 5771, "range": [ 14062, 14371 ], "source_revision": "d5776d5bdae66f74cdbd2aaeda0ef4ffb4148f08eeb21a72c16dd61bab582129", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "辛群(compact symplectic group)", "context": { "range": [ 14012, 14499 ], "quote": "因此，所有单位四元数在乘法下构成群 $\\mathrm{Sp}(1)$。对于更大的 $n$，还有称为[辛群(compact symplectic group)](../../../../../concepts/lie-theory/symplectic-group) $\\mathrm{Sp}(n)$ 的李群；它们由保持四元数范数平方的可逆四元数矩阵组成。由于这些群在量子力学中并不起重要作用，我们不再进一步研究。", "quote_sha256": "53da98588ebbedaedeafb3f2a730d06b9447ce21850eda683df877c893db70aa", "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-2bde0549a21915049cc995ff" }, { "concept_id": "urn:uuid:475077c6-8a0d-4954-9883-ba29330925d3", "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": "first-0f032f052560f8bd866d-en", "chapter": "6", "section": "6.2", "offset": 7208, "range": [ 14657, 14955 ], "source_revision": "525b7bc4b6973b05449bbbbcbdef1c067e7411218e20b20cb388b95687d1c4d8", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "symplectic groups", "context": { "range": [ 14531, 15195 ], "quote": "The unit quaternions form a group under multiplication, called $\\mathrm{Sp}(1)$. For larger $n$, there are Lie groups called [symplectic groups](../../../../../concepts/lie-theory/symplectic-group) $\\mathrm{Sp}(n)$; they consist of invertible quaternionic matrices that preserve the squared norm. They play no significant role in quantum mechanics, so we will not study them further.", "quote_sha256": "3e9eb4cb26f065058ff05e964ae3aa79dbc3e6ce9e9dcbb1b5f0047fa9b36d13", "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-67b23bea8ab0b8c6ecdfc386" } ], "registered_positions": [ { "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": "symplectic groups", "english": "symplectic groups", "offset": 5648 }, { "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": "辛群", "english": "compact symplectic group", "offset": 4367 } ], "statements": [], "resolved_references": [], "reading_order": [ "occ-2bde0549a21915049cc995ff", "occ-67b23bea8ab0b8c6ecdfc386" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 2, "located_occurrences": 2, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "86627c5dde10e279039aed318e1a980bcf820255fe1570e947b6a3e74eca122e", "optional_enrichments": [ { "id": "archived-475077c6-8a0d-4954-9883-ba29330925d3", "kind": "archived-composition", "actor": { "model": "gpt-6-luna", "reasoning_effort": "high" }, "source_status": "current", "stale_reasons": [], "review_status": { "generation": "complete", "structure": "validated", "semantic_review": "independent-luna-reviewed; human-review-not-performed", "translation_review": "not-performed; inherited-source-status" }, "result": { "overview_claims": [ { "id": "quaternionic-sp-n", "text": "Woit 在第6.2节将较大 $n$ 时的 $Sp(n)$ 描述为四元数元素可逆矩阵组成的李群，这些矩阵作用于四元数向量并保持四元数长度平方。", "evidence_ids": [ "e-2bfdec6ad5c23143e18b857c" ] }, { "id": "unit-quaternion-sp1", "text": "该节还指出，长度为 $1$ 的四元数在乘法下构成称为 $Sp(1)$ 的群，并将其与单位球面 $S^3\\subset\\mathbf R^4$ 联系起来。", "evidence_ids": [ "e-2bfdec6ad5c23143e18b857c" ] } ], "author_views": [ { "id": "woit-quaternionic-sp-view", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "summary": "Woit 先说明四元数长度平方 $|q|^2=q\\bar q$ 的乘法性质，再称单位四元数构成 $Sp(1)$。对于更大的 $n$，其概述是：$Sp(n)$ 由四元数项可逆矩阵组成，作用在四元数向量上并保持四元数长度平方。", "conditions": "$Sp(1)$ 对应长度为 $1$ 的四元数；对较大的 $n$，此处描述的是四元数矩阵及其保持长度平方的作用。", "notation": "本条目中的 $Sp(n)$ 按该节的四元数语境使用；该段并未展开更大 $n$ 时的矩阵条件。", "evidence_ids": [ "e-2bfdec6ad5c23143e18b857c" ] } ], "reading_suggestions": [ { "id": "read-woit-6-2", "text": "阅读 Woit 第6.2节中单位四元数与 $Sp(1)$ 的说明，再读关于较大 $n$ 时 $Sp(n)$ 的概述；后者只给出保持四元数长度平方的描述。", "target_ids": [ "e-2bfdec6ad5c23143e18b857c" ], "evidence_ids": [ "e-2bfdec6ad5c23143e18b857c" ] } ], "assessment": { "composition": "complete", "key_material": "available", "critical_gaps": [], "whole_book_coverage": "not-exhausted" }, "gaps": [ "Woit 第6.2节英文机器转录尚未人工核对；目前选段没有覆盖该群在全书中的其他用法。" ] }, "evidence": { "e-4adb4fdec29745927008b82e": { "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions", "anchor": "", "statement_id": null, "label": "Woit · §6.2", "quote_sha256": "38e804ca8bf72da0caa6a848fac738b6f955b8260e7a6c19e663a7bb54abe689" }, "e-2bfdec6ad5c23143e18b857c": { "route": "books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions", "anchor": "", "statement_id": null, "label": "Woit · §6.2", "quote_sha256": "0bce8a280ff15de23146eb82d5a78e663c125b1ab804f21620a6ed08b8a2ea4f" } }, "candidate_dependencies": [], "input_sha256": "a88d4659edcb2f400f0a845f4f1402ea4a9706a3ccebc4957a8badb034eea3cb", "archive_sha256": "c3e46eebe09d3554fd5b73239d1675c26ba46545b57cd36af757329671754850" } ] }