{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:5c0010c3-e567-45b1-926e-91dd0363f528", "labels": { "zh-CN": "李超括号", "en": "Lie superbracket" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "2b725283b81735034bf4404eae7d2bcbc63f86d24b2b962fb10101e9e8871af1", "occurrences": [ { "concept_id": "urn:uuid:5c0010c3-e567-45b1-926e-91dd0363f528", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket", "anchor": "first-7c533cddbd9d2f52b76c-zh", "chapter": "30", "section": "30.2", "offset": 3946, "range": [ 11748, 12051 ], "source_revision": "a48188b3c880cb3cdd0d6c3ef4904bbe64ddb27a96760877b32a6b652a0f69cc", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "李超括号(Lie superbracket)", "context": { "range": [ 11335, 12706 ], "quote": "\n\n> [!definition] 定义 · 李超代数\n>\n> 实或复向量空间 $V$ 上的[李超代数(Lie superalgebra)](../../../../../concepts/lie-theory/lie-superalgebra)结构由双线性映射[李超括号(Lie superbracket)](../../../../../concepts/lie-theory/lie-superbracket) $[\\cdot,\\cdot]_\\pm$ 给出。它在通常的或费米子的生成元 $X,Y,Z$ 上满足\n>\n> $$\n> [X,Y]_\\pm=-(-1)^{|X||Y|}[Y,X]_\\pm\n> $$\n>\n> 以及[超雅可比恒等式(super-Jacobi identity)](../../../../../concepts/lie-theory/super-jacobi-identity)\n>\n> $$\n> [X,[Y,Z]_\\pm]_\\pm=[[X,Y]_\\pm,Z]_\\pm+(-1)^{|X||Y|}[Y,[X,Z]_\\pm]_\\pm\n> $$\n>\n> 其中，通常生成元的 $|X|=0$，费米子生成元的 $|X|=1$。\n", "quote_sha256": "461171878c58aa20d2e238d67c3f0bf7b5933ec1ec473066f8eb5a7b4e20168b", "environment": "definition", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-c54f78bdca5a45d91c765979" }, { "concept_id": "urn:uuid:5c0010c3-e567-45b1-926e-91dd0363f528", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket", "anchor": "first-7c533cddbd9d2f52b76c-en", "chapter": "30", "section": "30.2", "offset": 6471, "range": [ 13428, 13725 ], "source_revision": "f741b6c43d1c81c0662bf299d14c14dac1ed68fd550ede08069c773abe9b83a4", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "Lie superbracket", "context": { "range": [ 12993, 14458 ], "quote": "\n\n> [!definition] Definition · Lie superalgebra\n>\n> A [Lie superalgebra](../../../../../concepts/lie-theory/lie-superalgebra) structure on a real or complex vector space $V$ is given by a [Lie superbracket](../../../../../concepts/lie-theory/lie-superbracket) $[\\cdot,\\cdot]_\\pm$, a bilinear map satisfying, on usual or fermionic generators $X,Y,Z$,\n>\n> $$\n> [X,Y]_\\pm=-(-1)^{|X||Y|}[Y,X]_\\pm\n> $$\n>\n> and the [super-Jacobi identity](../../../../../concepts/lie-theory/super-jacobi-identity)\n>\n> $$\n> [X,[Y,Z]_\\pm]_\\pm=[[X,Y]_\\pm,Z]_\\pm+(-1)^{|X||Y|}[Y,[X,Z]_\\pm]_\\pm\n> $$\n>\n> where $|X|=0$ for a usual generator and $|X|=1$ for a fermionic generator.\n", "quote_sha256": "57a8b3f1cc1dd7bb141414e46b8b40cbadac3eefb1599da14dc9ca348316d806", "environment": "definition", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-2a56277551fc7fd7aa85febc" } ], "registered_positions": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket", "lang": "en", "section": "30.2", "signal": "term-mark", "label": "Lie superbracket", "english": "Lie superbracket", "offset": 6053 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket", "lang": "zh-CN", "section": "30.2", "signal": "term-mark", "label": "李超括号", "english": "Lie superbracket", "offset": 3493 } ], "statements": [], "resolved_references": [], "reading_order": [ "occ-c54f78bdca5a45d91c765979", "occ-2a56277551fc7fd7aa85febc" ], "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": "ab06016e5a40e061cf4669e66981893735e1c8283eb1ac13541aa8a4db549357", "optional_enrichments": [] }