{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:d48ed7ff-3ca1-404d-9e8e-9c302f532e74", "labels": { "zh-CN": "辛形式", "en": "Symplectic form" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "415c6d081a288670c04f73b5e9eaf7321e6a873587166642a5e2652975f8ca42", "occurrences": [ { "concept_id": "urn:uuid:d48ed7ff-3ca1-404d-9e8e-9c302f532e74", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-14/03-symplectic-geometry", "anchor": "first-1128d1d3771008be8bd2-zh", "chapter": "14", "section": "14.3", "offset": 926, "range": [ 7471, 7888 ], "source_revision": "a5201f01c6f7e21eb1947b016df214d779bd89ce74c57dcc64ac23ec88adea46", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "definition", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "books/woit-qtgr/2025-10-20/zh/chapter-14/03-symplectic-geometry", "lang": "zh-CN", "section": "14.3", "signal": "named-definition", "label": "辛形式", "english": "", "offset": 856 } }, "label": "> [!definition] 定义（辛形式）", "context": { "range": [ 7471, 7888 ], "quote": "\n\n> [!definition] 定义（辛形式）\n> 向量空间 $V$ 上的辛形式 $\\omega$ 是一个双线性映射\n>\n> $$\n> \\omega:V\\times V\\to\\mathbf R\n> $$\n>\n> 并满足以下条件：\n>\n> - $\\omega$ 反对称，即 $\\omega(v,v^{\\prime})=-\\omega(v^{\\prime},v)$。\n> - $\\omega$ 非退化，即若 $v\\ne0$，则 $\\omega(v,\\cdot)\\in V^*$ 非零。带有辛形式 $\\omega$ 的向量空间 $V$ 称为辛向量空间。以辛形式取代内积而得到的几何称为辛几何。这里没有长度的概念，因为反对称性意味着 $\\omega(v,v)=0$。与正交群相对应的群称为辛群，由保持 $\\omega$ 的线性变换组成；我们将在[第 16 章](../chapter-16/index#section-16)详细研究它。\n", "quote_sha256": "8050babfe7a240eeac2a44fdc9bc380cb18e2ea76448d117df49e40e4ab98fad", "environment": "definition", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-14/03-symplectic-geometry", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-9c30d69e7e95979ba288bcd0" }, { "concept_id": "urn:uuid:d48ed7ff-3ca1-404d-9e8e-9c302f532e74", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-14/03-symplectic-geometry", "anchor": "first-0235925bab4334db2e44-en", "chapter": "14", "section": "14.3", "offset": 1734, "range": [ 7444, 8256 ], "source_revision": "806791b67ea3ad15ec69fdd6c3e219cda63631a7842713fdb19ce14eb503ab95", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "definition", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "books/woit-qtgr/2025-10-20/en/chapter-14/03-symplectic-geometry", "lang": "en", "section": "14.3", "signal": "named-definition", "label": "Symplectic form", "english": "Symplectic form", "offset": 1664 } }, "label": "> [!definition] Definition (Symplectic form)", "context": { "range": [ 7444, 8256 ], "quote": "\n\n> [!definition] Definition (Symplectic form)\n> A symplectic form $\\omega$ on a vector space $V$ is a bilinear map\n>\n> $$\n> \\omega:V\\times V\\to\\mathbf R\n> $$\n>\n> such that\n>\n> - $\\omega$ is antisymmetric: $\\omega(v,v^{\\prime})=-\\omega(v^{\\prime},v)$.\n> - $\\omega$ is nondegenerate: if $v\\ne0$, then $\\omega(v,\\cdot)\\in V^*$ is nonzero.A vector space $V$ with a symplectic form $\\omega$ is called a symplectic vector space. The geometry obtained by replacing an inner product with a symplectic form is called symplectic geometry. There is no notion of length in this geometry, since antisymmetry gives $\\omega(v,v)=0$. The symplectic group, analogous to the orthogonal group, consists of the linear transformations preserving $\\omega$; we will study it in detail in [chapter 16](../chapter-16/index#section-16).\n", "quote_sha256": "d6177ff23edc877a3066b760d4ef83c0944d2cd3f3ad479a0380647e0e334e45", "environment": "definition", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-14/03-symplectic-geometry", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-c8b3db8d3301e40e38b25adf" } ], "registered_positions": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-14/03-symplectic-geometry", "lang": "en", "section": "14.3", "signal": "named-definition", "label": "Symplectic form", "english": "Symplectic form", "offset": 1664 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-14/03-symplectic-geometry", "lang": "zh-CN", "section": "14.3", "signal": "named-definition", "label": "辛形式", "english": "", "offset": 856 } ], "statements": [], "resolved_references": [ { "raw": "[第 16 章](../chapter-16/index#section-16)", "href": "../chapter-16/index#section-16", "range": [ 7841, 7881 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/zh/chapter-16/index", "target_anchor": "section-16", "resolved": true, "reason": null, "source_revision": "a5201f01c6f7e21eb1947b016df214d779bd89ce74c57dcc64ac23ec88adea46", "source_route": "books/woit-qtgr/2025-10-20/zh/chapter-14/03-symplectic-geometry", "occurrence_id": "occ-9c30d69e7e95979ba288bcd0" }, { "raw": "[chapter 16](../chapter-16/index#section-16)", "href": "../chapter-16/index#section-16", "range": [ 8210, 8254 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/en/chapter-16/index", "target_anchor": "section-16", "resolved": true, "reason": null, "source_revision": "806791b67ea3ad15ec69fdd6c3e219cda63631a7842713fdb19ce14eb503ab95", "source_route": "books/woit-qtgr/2025-10-20/en/chapter-14/03-symplectic-geometry", "occurrence_id": "occ-c8b3db8d3301e40e38b25adf" } ], "reading_order": [ "occ-9c30d69e7e95979ba288bcd0", "occ-c8b3db8d3301e40e38b25adf" ], "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": "39aeafad72072d8ff7f10a116db3f272c9183f9be14366469b2b76973b5d0ce0", "optional_enrichments": [] }