Topological components of spaces of commuting elements in connected nilpotent Lie groups Article Swipe

PDF

Omar Antolín Camarena , Bernardo Villarreal ·

YOU? · · 2024 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2405.09652

We study the homotopy type of spaces of commuting elements in connected nilpotent Lie groups, via almost commuting elements in their Lie algebras. We give a necessary and sufficient condition on the fundamental group of such a Lie group $G$ to ensure $\mathrm{Hom}(\mathbb{Z}^k,G)$ is path-connected. In particular for the reduced upper unitriangular groups and the reduced generalized Heisenberg groups, $\mathrm{Hom}(\mathbb{Z}^k,G)$ is not path-connected, and we compute the homotopy type of its path-connected components in terms of Stiefel manifolds and the maximal torus of $G$.

Related Topics

Mathematics

Geography

Combinatorics

Concepts

Nilpotent Lie group Topology (electrical circuits) Mathematics Pure mathematics Lie theory Geography Combinatorics Lie conformal algebra Adjoint representation of a Lie algebra

Metadata

Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2405.09652
PDF: https://arxiv.org/pdf/2405.09652
OA Status: green
Related Works: 10
OpenAlex ID: https://openalex.org/W4397028171

All OpenAlex metadata

Raw OpenAlex JSON

OpenAlex ID: https://openalex.org/W4397028171

Canonical identifier for this work in OpenAlex
DOI: https://doi.org/10.48550/arxiv.2405.09652

Digital Object Identifier
Title: Topological components of spaces of commuting elements in connected nilpotent Lie groups

Work title
Type: preprint

OpenAlex work type
Language: en

Primary language
Publication year: 2024

Year of publication
Publication date: 2024-05-15

Full publication date if available
Authors: Omar Antolín Camarena, Bernardo Villarreal

List of authors in order
Landing page: https://arxiv.org/abs/2405.09652

Publisher landing page
PDF URL: https://arxiv.org/pdf/2405.09652

Direct link to full text PDF
Open access: Yes

Whether a free full text is available
OA status: green

Open access status per OpenAlex
OA URL: https://arxiv.org/pdf/2405.09652

Direct OA link when available
Concepts: Nilpotent, Lie group, Topology (electrical circuits), Mathematics, Pure mathematics, Lie theory, Geography, Combinatorics, Lie conformal algebra, Adjoint representation of a Lie algebra

Top concepts (fields/topics) attached by OpenAlex
Cited by: 0

Total citation count in OpenAlex
Related works (count): 10

Other works algorithmically related by OpenAlex

Full payload

id	https://openalex.org/W4397028171
doi	https://doi.org/10.48550/arxiv.2405.09652
ids.doi	https://doi.org/10.48550/arxiv.2405.09652
ids.openalex	https://openalex.org/W4397028171
fwci
type	preprint
title	Topological components of spaces of commuting elements in connected nilpotent Lie groups
biblio.issue
biblio.volume
biblio.last_page
biblio.first_page
topics[0].id	https://openalex.org/T10849
topics[0].field.id	https://openalex.org/fields/26
topics[0].field.display_name	Mathematics
topics[0].score	0.9171000123023987
topics[0].domain.id	https://openalex.org/domains/3
topics[0].domain.display_name	Physical Sciences
topics[0].subfield.id	https://openalex.org/subfields/2607
topics[0].subfield.display_name	Discrete Mathematics and Combinatorics
topics[0].display_name	Finite Group Theory Research
is_xpac	False
apc_list
apc_paid
concepts[0].id	https://openalex.org/C50555996
concepts[0].level	2
concepts[0].score	0.7020145654678345
concepts[0].wikidata	https://www.wikidata.org/wiki/Q840023
concepts[0].display_name	Nilpotent
concepts[1].id	https://openalex.org/C187915474
concepts[1].level	2
concepts[1].score	0.6043645143508911
concepts[1].wikidata	https://www.wikidata.org/wiki/Q622679
concepts[1].display_name	Lie group
concepts[2].id	https://openalex.org/C184720557
concepts[2].level	2
concepts[2].score	0.5394495129585266
concepts[2].wikidata	https://www.wikidata.org/wiki/Q7825049
concepts[2].display_name	Topology (electrical circuits)
concepts[3].id	https://openalex.org/C33923547
concepts[3].level	0
concepts[3].score	0.5360778570175171
concepts[3].wikidata	https://www.wikidata.org/wiki/Q395
concepts[3].display_name	Mathematics
concepts[4].id	https://openalex.org/C202444582
concepts[4].level	1
concepts[4].score	0.5198372006416321
concepts[4].wikidata	https://www.wikidata.org/wiki/Q837863
concepts[4].display_name	Pure mathematics
concepts[5].id	https://openalex.org/C143135876
concepts[5].level	5
concepts[5].score	0.4249183237552643
concepts[5].wikidata	https://www.wikidata.org/wiki/Q357300
concepts[5].display_name	Lie theory
concepts[6].id	https://openalex.org/C205649164
concepts[6].level	0
concepts[6].score	0.3605360984802246
concepts[6].wikidata	https://www.wikidata.org/wiki/Q1071
concepts[6].display_name	Geography
concepts[7].id	https://openalex.org/C114614502
concepts[7].level	1
concepts[7].score	0.17112082242965698
concepts[7].wikidata	https://www.wikidata.org/wiki/Q76592
concepts[7].display_name	Combinatorics
concepts[8].id	https://openalex.org/C73648015
concepts[8].level	3
concepts[8].score	0.06680256128311157
concepts[8].wikidata	https://www.wikidata.org/wiki/Q6543817
concepts[8].display_name	Lie conformal algebra
concepts[9].id	https://openalex.org/C203946495
concepts[9].level	4
concepts[9].score	0.0
concepts[9].wikidata	https://www.wikidata.org/wiki/Q1017106
concepts[9].display_name	Adjoint representation of a Lie algebra
keywords[0].id	https://openalex.org/keywords/nilpotent
keywords[0].score	0.7020145654678345
keywords[0].display_name	Nilpotent
keywords[1].id	https://openalex.org/keywords/lie-group
keywords[1].score	0.6043645143508911
keywords[1].display_name	Lie group
keywords[2].id	https://openalex.org/keywords/topology
keywords[2].score	0.5394495129585266
keywords[2].display_name	Topology (electrical circuits)
keywords[3].id	https://openalex.org/keywords/mathematics
keywords[3].score	0.5360778570175171
keywords[3].display_name	Mathematics
keywords[4].id	https://openalex.org/keywords/pure-mathematics
keywords[4].score	0.5198372006416321
keywords[4].display_name	Pure mathematics
keywords[5].id	https://openalex.org/keywords/lie-theory
keywords[5].score	0.4249183237552643
keywords[5].display_name	Lie theory
keywords[6].id	https://openalex.org/keywords/geography
keywords[6].score	0.3605360984802246
keywords[6].display_name	Geography
keywords[7].id	https://openalex.org/keywords/combinatorics
keywords[7].score	0.17112082242965698
keywords[7].display_name	Combinatorics
keywords[8].id	https://openalex.org/keywords/lie-conformal-algebra
keywords[8].score	0.06680256128311157
keywords[8].display_name	Lie conformal algebra
language	en
locations[0].id	pmh:oai:arXiv.org:2405.09652
locations[0].is_oa	True
locations[0].source.id	https://openalex.org/S4306400194
locations[0].source.issn
locations[0].source.type	repository
locations[0].source.is_oa	True
locations[0].source.issn_l
locations[0].source.is_core	False
locations[0].source.is_in_doaj	False
locations[0].source.display_name	arXiv (Cornell University)
locations[0].source.host_organization	https://openalex.org/I205783295
locations[0].source.host_organization_name	Cornell University
locations[0].source.host_organization_lineage	https://openalex.org/I205783295
locations[0].license
locations[0].pdf_url	https://arxiv.org/pdf/2405.09652
locations[0].version	submittedVersion
locations[0].raw_type
locations[0].license_id
locations[0].is_accepted	False
locations[0].is_published	False
locations[0].raw_source_name
locations[0].landing_page_url	http://arxiv.org/abs/2405.09652
locations[1].id	doi:10.48550/arxiv.2405.09652
locations[1].is_oa	True
locations[1].source.id	https://openalex.org/S4306400194
locations[1].source.issn
locations[1].source.type	repository
locations[1].source.is_oa	True
locations[1].source.issn_l
locations[1].source.is_core	False
locations[1].source.is_in_doaj	False
locations[1].source.display_name	arXiv (Cornell University)
locations[1].source.host_organization	https://openalex.org/I205783295
locations[1].source.host_organization_name	Cornell University
locations[1].source.host_organization_lineage	https://openalex.org/I205783295
locations[1].license
locations[1].pdf_url
locations[1].version
locations[1].raw_type	article
locations[1].license_id
locations[1].is_accepted	False
locations[1].is_published
locations[1].raw_source_name
locations[1].landing_page_url	https://doi.org/10.48550/arxiv.2405.09652
indexed_in	arxiv, datacite
authorships[0].author.id	https://openalex.org/A5076987222
authorships[0].author.orcid	https://orcid.org/0000-0002-5335-7471
authorships[0].author.display_name	Omar Antolín Camarena
authorships[0].author_position	first
authorships[0].raw_author_name	Antolín-Camarena, Omar
authorships[0].is_corresponding	False
authorships[1].author.id	https://openalex.org/A5011802109
authorships[1].author.orcid	https://orcid.org/0000-0001-6856-9509
authorships[1].author.display_name	Bernardo Villarreal
authorships[1].author_position	last
authorships[1].raw_author_name	Villarreal, Bernardo
authorships[1].is_corresponding	False
has_content.pdf	False
has_content.grobid_xml	False
is_paratext	False
open_access.is_oa	True
open_access.oa_url	https://arxiv.org/pdf/2405.09652
open_access.oa_status	green
open_access.any_repository_has_fulltext	False
created_date	2024-05-18T00:00:00
display_name	Topological components of spaces of commuting elements in connected nilpotent Lie groups
has_fulltext	False
is_retracted	False
updated_date	2025-11-06T06:51:31.235846
primary_topic.id	https://openalex.org/T10849
primary_topic.field.id	https://openalex.org/fields/26
primary_topic.field.display_name	Mathematics
primary_topic.score	0.9171000123023987
primary_topic.domain.id	https://openalex.org/domains/3
primary_topic.domain.display_name	Physical Sciences
primary_topic.subfield.id	https://openalex.org/subfields/2607
primary_topic.subfield.display_name	Discrete Mathematics and Combinatorics
primary_topic.display_name	Finite Group Theory Research
related_works	https://openalex.org/W2952605465, https://openalex.org/W3184611687, https://openalex.org/W2981695848, https://openalex.org/W4300900232, https://openalex.org/W4947501, https://openalex.org/W4386794951, https://openalex.org/W2256043873, https://openalex.org/W4320807737, https://openalex.org/W4293059793, https://openalex.org/W4230659680
cited_by_count	0
locations_count	2
best_oa_location.id	pmh:oai:arXiv.org:2405.09652
best_oa_location.is_oa	True
best_oa_location.source.id	https://openalex.org/S4306400194
best_oa_location.source.issn
best_oa_location.source.type	repository
best_oa_location.source.is_oa	True
best_oa_location.source.issn_l
best_oa_location.source.is_core	False
best_oa_location.source.is_in_doaj	False
best_oa_location.source.display_name	arXiv (Cornell University)
best_oa_location.source.host_organization	https://openalex.org/I205783295
best_oa_location.source.host_organization_name	Cornell University
best_oa_location.source.host_organization_lineage	https://openalex.org/I205783295
best_oa_location.license
best_oa_location.pdf_url	https://arxiv.org/pdf/2405.09652
best_oa_location.version	submittedVersion
best_oa_location.raw_type
best_oa_location.license_id
best_oa_location.is_accepted	False
best_oa_location.is_published	False
best_oa_location.raw_source_name
best_oa_location.landing_page_url	http://arxiv.org/abs/2405.09652
primary_location.id	pmh:oai:arXiv.org:2405.09652
primary_location.is_oa	True
primary_location.source.id	https://openalex.org/S4306400194
primary_location.source.issn
primary_location.source.type	repository
primary_location.source.is_oa	True
primary_location.source.issn_l
primary_location.source.is_core	False
primary_location.source.is_in_doaj	False
primary_location.source.display_name	arXiv (Cornell University)
primary_location.source.host_organization	https://openalex.org/I205783295
primary_location.source.host_organization_name	Cornell University
primary_location.source.host_organization_lineage	https://openalex.org/I205783295
primary_location.license
primary_location.pdf_url	https://arxiv.org/pdf/2405.09652
primary_location.version	submittedVersion
primary_location.raw_type
primary_location.license_id
primary_location.is_accepted	False
primary_location.is_published	False
primary_location.raw_source_name
primary_location.landing_page_url	http://arxiv.org/abs/2405.09652
publication_date	2024-05-15
publication_year	2024
referenced_works_count	0
abstract_inverted_index.a	25, 36
abstract_inverted_index.In	45
abstract_inverted_index.We	0, 23
abstract_inverted_index.in	10, 19, 73
abstract_inverted_index.is	43, 60
abstract_inverted_index.of	5, 7, 34, 69, 75, 82
abstract_inverted_index.on	30
abstract_inverted_index.to	40
abstract_inverted_index.we	64
abstract_inverted_index.$G$	39
abstract_inverted_index.Lie	13, 21, 37
abstract_inverted_index.and	27, 53, 63, 78
abstract_inverted_index.for	47
abstract_inverted_index.its	70
abstract_inverted_index.not	61
abstract_inverted_index.the	2, 31, 48, 54, 66, 79
abstract_inverted_index.via	15
abstract_inverted_index.$G$.	83
abstract_inverted_index.give	24
abstract_inverted_index.such	35
abstract_inverted_index.type	4, 68
abstract_inverted_index.group	33, 38
abstract_inverted_index.study	1
abstract_inverted_index.terms	74
abstract_inverted_index.their	20
abstract_inverted_index.torus	81
abstract_inverted_index.upper	50
abstract_inverted_index.almost	16
abstract_inverted_index.ensure	41
abstract_inverted_index.groups	52
abstract_inverted_index.spaces	6
abstract_inverted_index.Stiefel	76
abstract_inverted_index.compute	65
abstract_inverted_index.groups,	14, 58
abstract_inverted_index.maximal	80
abstract_inverted_index.reduced	49, 55
abstract_inverted_index.elements	9, 18
abstract_inverted_index.homotopy	3, 67
abstract_inverted_index.algebras.	22
abstract_inverted_index.commuting	8, 17
abstract_inverted_index.condition	29
abstract_inverted_index.connected	11
abstract_inverted_index.manifolds	77
abstract_inverted_index.necessary	26
abstract_inverted_index.nilpotent	12
abstract_inverted_index.Heisenberg	57
abstract_inverted_index.components	72
abstract_inverted_index.particular	46
abstract_inverted_index.sufficient	28
abstract_inverted_index.fundamental	32
abstract_inverted_index.generalized	56
abstract_inverted_index.unitriangular	51
abstract_inverted_index.path-connected	71
abstract_inverted_index.path-connected,	62
abstract_inverted_index.path-connected.	44
abstract_inverted_index.$\mathrm{Hom}(\mathbb{Z}^k,G)$	42, 59
cited_by_percentile_year
countries_distinct_count	0
institutions_distinct_count	2
citation_normalized_percentile