{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:7e2e6da6-b92e-45a5-b490-c11370b027b4", "labels": { "zh-CN": "结合代数", "en": "associative algebra" }, "alternative_labels": [ { "lang": "en", "value": "algebra" }, { "lang": "en", "value": "associative" }, { "lang": "zh-CN", "value": "代数" } ], "scope_note": "§28.1 calls an algebra a ring with a scalar-field vector-space structure.", "identity_status": "active", "identity_revision": "6b7a05bc75fe2f2d4741574b12ae4d95b280406392bce10d56d4343c39254b47", "occurrences": [ { "concept_id": "urn:uuid:7e2e6da6-b92e-45a5-b490-c11370b027b4", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-06/01-algebras-and-ideals", "anchor": "concept-6-associative-algebra", "chapter": "6", "section": "6.1", "offset": 1732, "range": [ 6488, 6732 ], "source_revision": "b14007c90d889a528ad167bc399531861f998294a0ab6687318a2d3cdc0b7deb", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "结合代数(associative algebra)", "context": { "range": [ 6208, 7035 ], "quote": "我们使用大写字母 $A$、$B$ 等来表示向量，因为代数最常出现在向量空间 $V$ 上的线性算符空间之中。然而，原则上没有什么能阻止使用更常见的记号 $u$、$v , . .$. 来表示向量，并写 $uv$ 表示它们的乘积。向量乘积一直用向量的简单并置来表示，但根据上下文，也可能出现其他记号，例如 $A \\times B , A \\otimes B , A \\wedge B$ 和 $[ A , B ]$。如果对一切 $A , B , C \\in { \\mathcal { A } }$. 都有 $A ( B C ) = ( A B ) C$，则称该代数为 结合代数(associative algebra)。如果对一切 $A$、$B \\in { \\mathcal { A } }$ 都有 $A B = B A$，则称其为 交换代数(commutative algebra)。", "quote_sha256": "844ecd0d91de52f7e8930fee87b333f81e572bf8de8136423288117a6ff4d6d9", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-06/01-algebras-and-ideals", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-cb9a523c57eeca34b01a8f6d" }, { "concept_id": "urn:uuid:7e2e6da6-b92e-45a5-b490-c11370b027b4", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-06/01-algebras-and-ideals", "anchor": "first-9f30c583afba7b647f72-en", "chapter": "6", "section": "6.1", "offset": 2291, "range": [ 6540, 6770 ], "source_revision": "2358bd7fdfb406dfc4a25849c3a46b00cea749852ae0264ca4ab53cc328ad69e", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "associative", "context": { "range": [ 6021, 7140 ], "quote": "We have used capital letters $A$, $B$, etc. to denote vectors because algebras most frequently arise in spaces of linear operators over a vector space $V$. There is, however, nothing in principle to prevent the more usual notation $u$, $v , . .$. for vectors and to write $uv$ for their product. The vector product has been denoted by a simple juxtaposition of vectors, but other notations such as $A \\times B , A \\otimes B , A \\wedge B$ and $[ A , B ]$ may arise, depending upon the context. The algebra is said to be associative if $A ( B C ) = ( A B ) C$ for all $A , B , C \\in { \\mathcal { A } }$. It is called commutative if $A B = B A$ for all $A$, $B \\in { \\mathcal { A } }$", "quote_sha256": "22cc43778bc6a0e9c0783a68ffc2f283d527591f2f44c7d51a166783ab6d0538", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-06/01-algebras-and-ideals", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-bd082730aa497fac7a15a41c" }, { "concept_id": "urn:uuid:7e2e6da6-b92e-45a5-b490-c11370b027b4", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras", "anchor": "first-33ec330b5f5555922192-zh", "chapter": "28", "section": "28.1", "offset": 571, "range": [ 8180, 8473 ], "source_revision": "d2a329a1c341b3d909aed292f756d80691f87ff3253e09de676b6b167a5fefb8", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "代数(algebra)", "context": { "range": [ 7823, 8618 ], "quote": "在数学中，“[环(ring)](../../../../../concepts/algebra/ring)”是带加法和乘法的集合，这些运算满足结合律与分配律，但不一定满足交换律；“[代数(algebra)](../../../../../concepts/algebra/algebra-over-a-field)”则是同时具有某个标量域上向量空间结构的环。正则对易关系与正则反对易关系分别定义了Weyl代数和Clifford代数。以复数为标量的情形最简单，我们先讨论复数情形，再讨论实数情形。", "quote_sha256": "06ca4caca9fac91edc206d023f1453a80ed150b88b9c0aa7f6e619ded5ac88a4", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-82225bc4eb6b7fdbb7fcca44" }, { "concept_id": "urn:uuid:7e2e6da6-b92e-45a5-b490-c11370b027b4", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras", "anchor": "first-33ec330b5f5555922192-en", "chapter": "28", "section": "28.1", "offset": 676, "range": [ 7386, 7675 ], "source_revision": "1437e8e29fe07ba5486788969d959b4b958fe09241e74dc8c02204e7032cf358", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "algebra", "context": { "range": [ 6944, 7964 ], "quote": "In mathematics, a “[ring](../../../../../concepts/algebra/ring)” is a set with associative and distributive addition and multiplication laws, not necessarily commutative. An “[algebra](../../../../../concepts/algebra/algebra-over-a-field)” is a ring that is also a vector space over a field of scalars. The canonical commutation and anticommutation relations define the Weyl and Clifford algebras, respectively. 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