{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:03eb86c3-86f8-40cb-8e28-d7d053422767", "labels": { "zh-CN": "闭子群", "en": "closed subgroup" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "ef083b742a23bbea6e4727221baa8921fa21e43593da3affa29c123d0ca9e86b", "occurrences": [ { "concept_id": "urn:uuid:03eb86c3-86f8-40cb-8e28-d7d053422767", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-10/08-topological-groups", "anchor": "first-0be35a567ad49081ee79-zh", "chapter": "10", "section": "10.8", "offset": 3876, "range": [ 8212, 8451 ], "source_revision": "40e39c23ad4632d9a97f04f151312b72ae74facf203e5c4311357aeafe0559dc", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "闭子群(closed subgroup)", "context": { "range": [ 7490, 8452 ], "quote": "群 $H$ 的子群 $G$ 连同其相对拓扑称为 $G$ 的 拓扑子群(topological subgroup)。为证明任意子群 $H$ 关于相对拓扑成为拓扑子群，设 $U ^ { \\prime } = H \\cap U$，其中 $U$ 是 $G .$ 的任意开子集。由映射 $\\phi$ 的连续性，对任意一对满足 $g h \\in U ^ { \\prime } \\subset U$ 的点 $g , h \\in H$，存在 $G$ 的开集 $A$ 和 $B$，使得 $A \\times B \\subset \\phi ^ { - 1 } ( U )$。由此推出 $\\phi ( A ^ { \\prime } \\times B ^ { \\prime } ) \\subset H \\cap U$，其中 $A ^ { \\prime } = A \\cap H$，$B ^ { \\prime } = B \\cap H$，并且 $\\left. \\phi \\right| _ { H }$ 的连续性是直接的。类似地，逆映射 $\\tau$ 限制在 $H$ 上时连续。若 $H$ 是 $G$ 中的闭集，则称其为 $G$ 的 闭子群(closed subgroup)。", "quote_sha256": "da288dafa542dfb428b6b09d832ad6c7a475636a27b8c5b883a4c346c589c501", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-10/08-topological-groups", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-b1abb9400da0e029a4aa2f45" }, { "concept_id": "urn:uuid:03eb86c3-86f8-40cb-8e28-d7d053422767", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-19/03-lie-subgroups", "anchor": "concept-19-closed-subgroup", "chapter": "19", "section": "19.3", "offset": 671, "range": [ 5400, 5636 ], "source_revision": "9eba7646d656cc81248004a9939877c26696aea87c347e90bd7234abe7b865f5", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "闭子群(closed subgroup)", "context": { "range": [ 5060, 5984 ], "quote": "李群 $G$ 的 李子群(Lie subgroup) $H$ 是一个子群，它本身是李群，并且使得由 $i ( g ) = g$ 定义的自然单射 $i : H \\to G$ 将其变为 $G$ 的嵌入子流形。如果此外 $H$ 是 $G$ 的闭子集，则称其为 闭子群(closed subgroup)。在这种情况下，嵌入是正则的，其拓扑是由 $G$ 的拓扑诱导的拓扑（见例 15.12）。单射 $i$ 诱导出李代数同态 $i _ { * } : { \\mathcal { H } } \\to ( G )$，它显然是 $\\mathcal{H}$ 与 $\\mathcal{G}$ 的一个李子代数的同构。因此，我们可以把该李子群的李代数视为 $\\mathcal{G}$ 的一个李子代数。", "quote_sha256": "e4e410e4332e5ea957c73edac57171da68c39e681a397391409c7bd3d62d2b1c", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-19/03-lie-subgroups", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-288b212fca19d853a7954bf6" }, { "concept_id": "urn:uuid:03eb86c3-86f8-40cb-8e28-d7d053422767", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-19/03-lie-subgroups", "anchor": "first-0be35a567ad49081ee79-en", "chapter": "19", "section": "19.3", "offset": 763, "range": [ 4986, 5220 ], "source_revision": "2f032245c9f12c9025411eb618055771e850f9b439b20768a9bd1a8a14520399", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "closed subgroup", "context": { "range": [ 4554, 5830 ], "quote": "A Lie subgroup $H$ of a Lie group $G$ is a subgroup that is a Lie group, and such that the natural injection map $i : H \\to G$ defined by $i ( g ) = g$ makes it into an embedded submanifold of $G$. It is called a closed subgroup if in addition $H$ is a closed subset of $G$. In this case the embedding is regular and its topology is that induced by the topology of $G$ (see Example 15.12). The injection $i$ induces a Lie algebra homomorphism $i _ { * } : { \\mathcal { H } } \\to ( G )$ , which is clearly an isomorphism of $\\mathcal{H}$ with a Lie subalgebra of $\\mathcal{G}$. We may therefore regard the Lie algebra of the Lie subgroup as being a Lie subalgebra of $\\mathcal{G}$.", "quote_sha256": "c50b876f43348f062b343f1fabaa301422cd282a8fd08f3ed7159c09695cc978", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-19/03-lie-subgroups", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-90db57e9208ce5b59d2c6ad7" } ], "registered_positions": [ { "route": "zh/chapter-10/08-topological-groups", "lang": "zh-CN", "section": "10.8", "signal": "term-mark", "label": "闭子群", "english": "closed subgroup", "offset": 3303 }, { "route": "en/chapter-19/03-lie-subgroups", "lang": "en", "section": "19.3", "signal": "term-mark", "label": "closed subgroup", "english": "closed subgroup", "offset": 572 }, { "route": "zh/chapter-19/03-lie-subgroups", "lang": "zh-CN", "section": "19.3", "signal": "term-mark", "label": "闭子群", "english": "closed subgroup", "offset": 515 } ], "statements": [], "resolved_references": [ { "raw": "href=\"../chapter-15/04-tangent-map-and-submanifolds#example-15-12\"", "href": "../chapter-15/04-tangent-map-and-submanifolds#example-15-12", "range": [ 5674, 5740 ], "relation": "references", "target_route": "zh/chapter-15/04-tangent-map-and-submanifolds", "target_anchor": "example-15-12", "resolved": true, "reason": null, "source_revision": "9eba7646d656cc81248004a9939877c26696aea87c347e90bd7234abe7b865f5", "source_route": "zh/chapter-19/03-lie-subgroups", "occurrence_id": "occ-288b212fca19d853a7954bf6" }, { "raw": "href=\"../chapter-15/04-tangent-map-and-submanifolds#example-15-12\"", "href": "../chapter-15/04-tangent-map-and-submanifolds#example-15-12", "range": [ 5369, 5435 ], "relation": "references", "target_route": "en/chapter-15/04-tangent-map-and-submanifolds", "target_anchor": "example-15-12", "resolved": true, "reason": null, "source_revision": "2f032245c9f12c9025411eb618055771e850f9b439b20768a9bd1a8a14520399", "source_route": "en/chapter-19/03-lie-subgroups", "occurrence_id": "occ-90db57e9208ce5b59d2c6ad7" } ], "reading_order": [ "occ-b1abb9400da0e029a4aa2f45", "occ-288b212fca19d853a7954bf6", "occ-90db57e9208ce5b59d2c6ad7" ], "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": "c92a343eca0c9cb7ec35acaebbaf7da11215c8f145fa2e54b3cd5b01a55adb73", "optional_enrichments": [] }