See Lie group on Wiktionary
Download JSON data for Lie group meaning in All languages combined (3.8kB)
{
"etymology_text": "Named for Norwegian mathematician Sophus Lie.",
"forms": [
{
"form": "Lie groups",
"tags": [
"plural"
]
}
],
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"expansion": "Lie group (plural Lie groups)",
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"lang": "English",
"lang_code": "en",
"pos": "noun",
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"name": "Manifolds",
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"kind": "topical",
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"name": "Topology",
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{
"text": "1994, Silvio Levy (translator), Albert S. Schwarz, Topology for Physicists, [1989, A. S. Shvarts, Kvantovaya teoriya polya i topologiya], Springer, 1996, 2nd Printing, page 233,\nEvery connected Lie group is homotopically equivalent to its maximal compact subgroup. This reduces the study of the homotopy and homology of Lie groups to the compact case."
},
{
"ref": "2009, Mikio Nakahara, Geometry, Topology and Physics, 2nd edition, Taylor & Francis, page 207",
"text": "A Lie group is a manifold on which the group manipulations, product and inverse, arc defined. Lie groups play an extremely important role in the theory of fibre bundles and also find vast applications in physics.",
"type": "quotation"
},
{
"ref": "2009, Boris Khesin, Robert Wendt, The Geometry of Infinite-Dimensional Groups, Springer, page 1",
"text": "As is well known, in finite dimensions each Lie group is, at least locally near the identity, completely described by its Lie algebra.",
"type": "quotation"
}
],
"glosses": [
"Any group that is a smooth manifold and whose group operations are differentiable."
],
"hypernyms": [
{
"word": "topological group"
}
],
"hyponyms": [
{
"word": "circle group"
},
{
"word": "Möbius group"
}
],
"id": "en-Lie_group-en-noun-wYkNtTZP",
"links": [
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"raw_glosses": [
"(topology) Any group that is a smooth manifold and whose group operations are differentiable."
],
"related": [
{
"word": "group of Lie type"
},
{
"word": "Lie algebra"
}
],
"topics": [
"mathematics",
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"translations": [
{
"code": "da",
"lang": "Danish",
"sense": "analytic group that is also a smooth manifold",
"tags": [
"common-gender"
],
"word": "Liegruppe"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "analytic group that is also a smooth manifold",
"word": "Lien ryhmä"
},
{
"code": "fr",
"lang": "French",
"sense": "analytic group that is also a smooth manifold",
"tags": [
"masculine"
],
"word": "groupe de Lie"
},
{
"code": "de",
"lang": "German",
"sense": "analytic group that is also a smooth manifold",
"tags": [
"feminine"
],
"word": "Lie-Gruppe"
},
{
"code": "de",
"lang": "German",
"sense": "analytic group that is also a smooth manifold",
"tags": [
"feminine"
],
"word": "Liesche Gruppe"
},
{
"code": "it",
"lang": "Italian",
"sense": "analytic group that is also a smooth manifold",
"tags": [
"masculine"
],
"word": "gruppo di Lie"
},
{
"code": "ja",
"lang": "Japanese",
"roman": "rīgun",
"sense": "analytic group that is also a smooth manifold",
"word": "リー群"
},
{
"code": "es",
"lang": "Spanish",
"sense": "analytic group that is also a smooth manifold",
"tags": [
"masculine"
],
"word": "grupo de Lie"
}
],
"wikipedia": [
"Lie group",
"Sophus Lie"
]
}
],
"word": "Lie group"
}
{
"etymology_text": "Named for Norwegian mathematician Sophus Lie.",
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"word": "circle group"
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"word": "Möbius group"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
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"word": "group of Lie type"
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"word": "Lie algebra"
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{
"text": "1994, Silvio Levy (translator), Albert S. Schwarz, Topology for Physicists, [1989, A. S. Shvarts, Kvantovaya teoriya polya i topologiya], Springer, 1996, 2nd Printing, page 233,\nEvery connected Lie group is homotopically equivalent to its maximal compact subgroup. This reduces the study of the homotopy and homology of Lie groups to the compact case."
},
{
"ref": "2009, Mikio Nakahara, Geometry, Topology and Physics, 2nd edition, Taylor & Francis, page 207",
"text": "A Lie group is a manifold on which the group manipulations, product and inverse, arc defined. Lie groups play an extremely important role in the theory of fibre bundles and also find vast applications in physics.",
"type": "quotation"
},
{
"ref": "2009, Boris Khesin, Robert Wendt, The Geometry of Infinite-Dimensional Groups, Springer, page 1",
"text": "As is well known, in finite dimensions each Lie group is, at least locally near the identity, completely described by its Lie algebra.",
"type": "quotation"
}
],
"glosses": [
"Any group that is a smooth manifold and whose group operations are differentiable."
],
"links": [
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"topology",
"topology"
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"smooth manifold",
"smooth manifold"
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"raw_glosses": [
"(topology) Any group that is a smooth manifold and whose group operations are differentiable."
],
"topics": [
"mathematics",
"sciences",
"topology"
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"wikipedia": [
"Lie group",
"Sophus Lie"
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],
"translations": [
{
"code": "da",
"lang": "Danish",
"sense": "analytic group that is also a smooth manifold",
"tags": [
"common-gender"
],
"word": "Liegruppe"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "analytic group that is also a smooth manifold",
"word": "Lien ryhmä"
},
{
"code": "fr",
"lang": "French",
"sense": "analytic group that is also a smooth manifold",
"tags": [
"masculine"
],
"word": "groupe de Lie"
},
{
"code": "de",
"lang": "German",
"sense": "analytic group that is also a smooth manifold",
"tags": [
"feminine"
],
"word": "Lie-Gruppe"
},
{
"code": "de",
"lang": "German",
"sense": "analytic group that is also a smooth manifold",
"tags": [
"feminine"
],
"word": "Liesche Gruppe"
},
{
"code": "it",
"lang": "Italian",
"sense": "analytic group that is also a smooth manifold",
"tags": [
"masculine"
],
"word": "gruppo di Lie"
},
{
"code": "ja",
"lang": "Japanese",
"roman": "rīgun",
"sense": "analytic group that is also a smooth manifold",
"word": "リー群"
},
{
"code": "es",
"lang": "Spanish",
"sense": "analytic group that is also a smooth manifold",
"tags": [
"masculine"
],
"word": "grupo de Lie"
}
],
"word": "Lie group"
}
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-05-09 from the enwiktionary dump dated 2024-05-02 using wiktextract (4d5d0bb and edd475d). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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