A finite presentation of universal coverings of lie tori

Saeid Azam; Hiroyuki Yamane; Malihe Yousofzadeh

doi:10.2977/PRIMS/16

A finite presentation of universal coverings of lie tori

Saeid Azam^*, Hiroyuki Yamane, Malihe Yousofzadeh

^*この論文の責任著者

数学科

研究成果: ジャーナルへの寄稿 › 学術論文 › 査読

3 被引用数 (Scopus)

抄録

Using the well-known recognition and structural theorem(s) for root-graded Lie algebras and their universal coverings, we give a finite presentation for the universal covering algebra of a centerless Lie torus of type X ≠ A, C, BC. We follow a unified approach for the types under consideration.

本文言語	英語
ページ（範囲）	507-548
ページ数	42
ジャーナル	Publications of the Research Institute for Mathematical Sciences
巻	46
号	3
DOI	https://doi.org/10.2977/PRIMS/16
出版ステータス	出版済み - 2010

ASJC Scopus 主題領域

数学一般

文献へのアクセス

10.2977/PRIMS/16

引用スタイル

@article{0b64f8f824b2485cbcf3a7092c13e85d,

title = "A finite presentation of universal coverings of lie tori",

abstract = "Using the well-known recognition and structural theorem(s) for root-graded Lie algebras and their universal coverings, we give a finite presentation for the universal covering algebra of a centerless Lie torus of type X ≠ A, C, BC. We follow a unified approach for the types under consideration.",

keywords = "Extended affine lie algebras, Finite presentation, Lie tori, Root graded Lie algebras, Universal enveloping algebra",

author = "Saeid Azam and Hiroyuki Yamane and Malihe Yousofzadeh",

note = "Funding Information: Research reported in this publication was supported by the Bill & Melinda Gates Foundation, the National Institute on Aging of the National Institutes of Health (award P30AG047845), and the National Institute of Mental Health of the National Institutes of Health (award R01MH110163). The content is solely the responsibility of the authors and does not necessarily represent the official views of the Bill & Melinda Gates Foundation or the National Institutes of Health. We also thank the countless individuals who have contributed to GBD 2016 and Future Health Scenarios research team in various capacities. Funding Information: Laith J Abu-Raddad acknowledges the support of Qatar National Research Fund (NPRP 9-040-3-008) who provided the main funding for generating the data provided to the GBD effort. Anurag Agrawal acknowledges the Wellcome Trust DBT India Alliance fellowship. Ashish Awasthi received financial support from Department of Science and Technology, Government of India through INSPIRE Faculty program. Alaa Badawi acknowledges the Public Health Agency of Canada. The scientific work of Aleksandra Barac is part of the Project No. III45005 granted by Ministry of Education, Science and Technological Development of the Republic of Serbia. Till B{\"a}rnighausen is funded by the Alexander von Humboldt Foundation through the Alexander von Humboldt Professorship endowed by the German Federal Ministry of Education and Research; he is also supported by the Wellcome Trust, the European Commission, the Clinton Health Access Initiative and NICHD of NIH [R01-HD084233], NIAID of NIH [R01-AI124389 and R01-AI112339] and FIC of NIH [D43-TW009775]. Dr Yannick B{\'e}jot reports grants and personal fees from AstraZeneca and Boehringer-Ingelheim, and personal fees from Daiichi-Sankyo, Pfizer-BMS, MSD, Bayer, and Covidiem, outside the submitted work. Boris Bikbov has received funding from the European Union's Horizon 2020 research and innovation programme under Marie Sklodowska-Curie grant agreement No. 703226. Boris Bikbov acknowledges that work related to this paper has been done on the behalf of the GBD Genitourinary Disease Expert Group. Cyrus Cooper reports consultancy fees from Alliance for Better Bond Health, Amgen, Eli Lilly, GlaxoSmithKline, Medtronic, Merck, Novartis, Pfizer, Roche, Servier, Takeda, and UCB. Jos{\'e} das Neves was supported in his contribution to this work by a Fellowship from Funda{\c c}{\~a}o para a Ci{\^e}ncia e a Tecnologia, Portugal (SFRH/BPD/92934/2013). Barbora de Courten is supported by National Heart Foundation Future Leader Fellowship (100864). Kebede Deribe is funded by a Wellcome Trust Intermediate Fellowship in Public Health and Tropical Medicine, grant number 201900. Mir Sohail Fazeli reports personal fees from Doctor Evidence LLC, outside the submitted work. Joao Fernandes is supported by FCT - Funda{\c c}{\~a}o para a Ci{\^e}ncia e a Tecnologia (Grant number UID/Multi/50016/2013). Katherine Gibney is supported by an Australian National Health and Medical Research Council (NHMRC) early career fellowship. Amador Goodridge acknowledges Sistema Nacional de Investigaci{\'o}n (SNI) de Panam{\'a} & Secretar{\'i}a Nacional de Ciencia, Tecnolog{\'i}a e Innovaci{\'o}n (SENACYT). Simon I Hay is funded by grants from the Bill & Melinda Gates Foundation (OPP1106023, OPP1119467, OPP1093011, and OPP1132415). Manami Inoue was the beneficiary of a financial contribution from the AXA Research Fund as chair-holder of the AXA Department of Health and Human Security, Graduate School of Medicine, The University of Tokyo. The AXA Research Fund had no role in this work. Shariful Islam has received a postdoctoral research fellowship from the George Institute for Global Health and career transition grants from High Blood Pressure Research Council of Australia. Ministry of Education Science and Technological Development of the Republic of Serbia has co-financed the Serbian part of Mihajlo Jakovljevic's GBD-related contribution through Grant OI 175 014; publication of results was not contingent upon the Ministry's approval. Panniyammakal Jeemon received clinical and public health intermediate fellowship from the Wellcome Trust and Department of Biotechnology, India Alliance (2015–2020). Nicholas Kassebaum reports personal fees and non-financial support from Vifor Pharmaceuticals, outside the submitted work. S Vittal Katikireddi was funded by a NRS Scottish Senior Clinical Fellowship (SCAF/15/02), the UK Medical Research Council (MC_UU_12017/13 & MC_UU_12017/15) and the Scottish Government Chief Scientist Office (SPHSU13 & SPHSU15). Christian Kieling has received support from Brazilian governmental research funding agencies Conselho Nacional de Desenvolvimento Cient{\'i}fico e Tecnol{\'o}gico (CNPq), Coordena{\c c}{\~a}o de Aperfei{\c c}oamento de Pessoal de N{\'i}vel Superior (CAPES), Funda{\c c}{\~a}o de Amparo {\`a} Pesquisa do Estado do Rio Grande do Sul (Fapergs), and Hospital de Cl{\'i}nicas de Porto Alegre (FIPE/HCPA). Ai Koyanagi's work was supported by the Miguel Servet contract financed by the CP13/00150 and PI15/00862 projects, integrated into the National R + D + I and funded by the ISCIII - General Branch Evaluation and Promotion of Health Research - and the European Regional Development Fund (ERDF-FEDER). Jeffrey Lazarus reports grants and personal fees from AbbVie, Gilead Sciences, and MSD, outside the submitted work. Katharine J Looker thanks the National Institute for Health Research Health Protection Research Unit (NIHR HPRU) in Evaluation of Interventions at the University of Bristol, in partnership with Public Health England (PHE), for research support. Katharine J Looker received separate funding from the World Health Organization and Sexual Health 24 during the course of this study. These funders had no role in the writing of the manuscript nor the decision to submit it for publication. The views expressed are those of the authors and not necessarily those of the National Health Service, the NIHR, the Department of Health or Public Health England. Azeem Majid and Imperial College London are grateful for support from the NW London NIHR Collaboration for Leadership in Applied Health Research & Care. Francisco Martins-Melo acknowledges a postdoctoral fellowship from the CAPES (Brazilian Federal Agency for Support and Evaluation of Graduate Education), outside the submitted work. Mohsen Mazidi was supported by the World Academy of Sciences and Chinese Academy of Sciences. John McGrath received John Cade Fellowship APP1056929 from the National Health and Medical Research Council, and Niels Bohr Professorship from the Danish National Research Foundation. Under contract, Ted Miller is leading an independent Measurement and Evaluation of the AB InBev Global Smart Drinking Goals Initiative, which aims to reduce the harmful use of alcohol. The baseline harm levels used in that evaluation will build from GBD's harm estimates. Philip Mitchell's research is supported by an Australian NHMRC Program Grant (no. 1037196). Ulrich Mueller gratefully acknowledges financial Support from the German National Cohort Study (BMBF grant # 01ER1511/D). Olanrewaju Oladimeji is a Senior Research Specialist at the Human Sciences Research Council (HSRC) and Doctoral Candidate at the University of KwaZulu-Natal (UKZN), South Africa; we acknowledge the institutional supports from HSRC and UKZN for him to participate in this study. Alberto Ortiz was supported by the Spanish Government (Intensificacion ISCIIII FEDER funds and RETIC REDINREN RD016/0019). Mayowa Owolabi is supported by U54HG007479 from the NIH. Norberto Perico would like to acknowledge that the work related to this paper has been done on behalf of the GBD Genitourinary Disease Expert Group. Giuseppe Remuzzi acknowledges that the work related to this paper has been done on behalf of the GBD Genitourinary Diseases Expert Group supported by the International Society of Nephrology (ISN). Luz Myriam Reynales-Shigematsu acknowledges the Global Adult Tobacco Survey, GATS Mexico 2015, with financial support provided by the CONADIC, Ministry of Health, Mexico, and the Bloomberg Initiative to Reduce Tobacco Use through the CDC Foundation with a grant from Bloomberg Philanthropies. Aletta E Schutte received support from the South African Medical Research Council and the National Research Foundation's SARChI Programme. Mark Shrime reports grants from the GE Foundation and the the Damon Runyon Cancer Research Foundation GE Safe Surgery 2020 Project. Jasvinder Singh reports consultancy fees from Regeneron, Merz, Iroko, Bioiberica, Crealta/Horizon, Allergan, UBM LLC, WebMD, and the American College of Rheumatology, and consultancy fees and grants from Savient, Takeda. Jasvinder Singh serves as the principal investigator for an investigator-initiated study funded by Horizon pharmaceuticals through a grant to DINORA, Inc, a 501c3 entity; he is also on the steering committee of OMERACT, an international organisation that develops measures for clinical trials and receives arms length funding from 36 pharmaceutical companies. Michael Soljak acknowledges funding from Public Health England for modelling of NCD prevalence. Cassandra Szoeke acknowledges the NHMRC as well as grants from Lundbeck, and Alzheimer's Association, outside the submitted work. In addition, Cassandra Szoeke has a patent, PCT/AU2008/001556 issued. Rafael Tabar{\'e}s-Seisdedos was supported in part by grant PROMETEOII/2015/021 from Generalitat Valenciana and the national grants PI14/00894 and PIE14/00031 from ISCIII-FEDER. Amand Thrift was supported by a Fellowship from the National Health & Medical Research Council (Australia; 1042600). Stefano Tyrovola's work was supported by the Foundation for Education and European Culture (IPEP), the Sara Borrell postdoctoral programme (reference no. CD15/00019 from the Instituto de Salud Carlos III [ISCIII – Spain]) and the Fondos Europeo de Desarrollo Regional (FEDER). Job van Boven received support from the Department of Clinical Pharmacy and Clinical Pharmacology, University Medical Center Groningen, University of Groningen, Netherlands. Lijing Yan is partially supported by the National Natural Sciences Foundation of China grants (71233001 and 71490732). Marcel Yotebieng is partially supported by the NIAID U01AI096299 and the NICHD R01HD087993.",

year = "2010",

doi = "10.2977/PRIMS/16",

language = "英語",

volume = "46",

pages = "507--548",

journal = "Publications of the Research Institute for Mathematical Sciences",

issn = "0034-5318",

publisher = "European Mathematical Society Publishing House",

number = "3",

}

TY - JOUR

T1 - A finite presentation of universal coverings of lie tori

AU - Azam, Saeid

AU - Yamane, Hiroyuki

AU - Yousofzadeh, Malihe

N1 - Funding Information: Research reported in this publication was supported by the Bill & Melinda Gates Foundation, the National Institute on Aging of the National Institutes of Health (award P30AG047845), and the National Institute of Mental Health of the National Institutes of Health (award R01MH110163). The content is solely the responsibility of the authors and does not necessarily represent the official views of the Bill & Melinda Gates Foundation or the National Institutes of Health. We also thank the countless individuals who have contributed to GBD 2016 and Future Health Scenarios research team in various capacities. Funding Information: Laith J Abu-Raddad acknowledges the support of Qatar National Research Fund (NPRP 9-040-3-008) who provided the main funding for generating the data provided to the GBD effort. Anurag Agrawal acknowledges the Wellcome Trust DBT India Alliance fellowship. Ashish Awasthi received financial support from Department of Science and Technology, Government of India through INSPIRE Faculty program. Alaa Badawi acknowledges the Public Health Agency of Canada. The scientific work of Aleksandra Barac is part of the Project No. III45005 granted by Ministry of Education, Science and Technological Development of the Republic of Serbia. Till Bärnighausen is funded by the Alexander von Humboldt Foundation through the Alexander von Humboldt Professorship endowed by the German Federal Ministry of Education and Research; he is also supported by the Wellcome Trust, the European Commission, the Clinton Health Access Initiative and NICHD of NIH [R01-HD084233], NIAID of NIH [R01-AI124389 and R01-AI112339] and FIC of NIH [D43-TW009775]. Dr Yannick Béjot reports grants and personal fees from AstraZeneca and Boehringer-Ingelheim, and personal fees from Daiichi-Sankyo, Pfizer-BMS, MSD, Bayer, and Covidiem, outside the submitted work. Boris Bikbov has received funding from the European Union's Horizon 2020 research and innovation programme under Marie Sklodowska-Curie grant agreement No. 703226. Boris Bikbov acknowledges that work related to this paper has been done on the behalf of the GBD Genitourinary Disease Expert Group. Cyrus Cooper reports consultancy fees from Alliance for Better Bond Health, Amgen, Eli Lilly, GlaxoSmithKline, Medtronic, Merck, Novartis, Pfizer, Roche, Servier, Takeda, and UCB. José das Neves was supported in his contribution to this work by a Fellowship from Fundação para a Ciência e a Tecnologia, Portugal (SFRH/BPD/92934/2013). Barbora de Courten is supported by National Heart Foundation Future Leader Fellowship (100864). Kebede Deribe is funded by a Wellcome Trust Intermediate Fellowship in Public Health and Tropical Medicine, grant number 201900. Mir Sohail Fazeli reports personal fees from Doctor Evidence LLC, outside the submitted work. Joao Fernandes is supported by FCT - Fundação para a Ciência e a Tecnologia (Grant number UID/Multi/50016/2013). Katherine Gibney is supported by an Australian National Health and Medical Research Council (NHMRC) early career fellowship. Amador Goodridge acknowledges Sistema Nacional de Investigación (SNI) de Panamá & Secretaría Nacional de Ciencia, Tecnología e Innovación (SENACYT). Simon I Hay is funded by grants from the Bill & Melinda Gates Foundation (OPP1106023, OPP1119467, OPP1093011, and OPP1132415). Manami Inoue was the beneficiary of a financial contribution from the AXA Research Fund as chair-holder of the AXA Department of Health and Human Security, Graduate School of Medicine, The University of Tokyo. The AXA Research Fund had no role in this work. Shariful Islam has received a postdoctoral research fellowship from the George Institute for Global Health and career transition grants from High Blood Pressure Research Council of Australia. Ministry of Education Science and Technological Development of the Republic of Serbia has co-financed the Serbian part of Mihajlo Jakovljevic's GBD-related contribution through Grant OI 175 014; publication of results was not contingent upon the Ministry's approval. Panniyammakal Jeemon received clinical and public health intermediate fellowship from the Wellcome Trust and Department of Biotechnology, India Alliance (2015–2020). Nicholas Kassebaum reports personal fees and non-financial support from Vifor Pharmaceuticals, outside the submitted work. S Vittal Katikireddi was funded by a NRS Scottish Senior Clinical Fellowship (SCAF/15/02), the UK Medical Research Council (MC_UU_12017/13 & MC_UU_12017/15) and the Scottish Government Chief Scientist Office (SPHSU13 & SPHSU15). Christian Kieling has received support from Brazilian governmental research funding agencies Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul (Fapergs), and Hospital de Clínicas de Porto Alegre (FIPE/HCPA). Ai Koyanagi's work was supported by the Miguel Servet contract financed by the CP13/00150 and PI15/00862 projects, integrated into the National R + D + I and funded by the ISCIII - General Branch Evaluation and Promotion of Health Research - and the European Regional Development Fund (ERDF-FEDER). Jeffrey Lazarus reports grants and personal fees from AbbVie, Gilead Sciences, and MSD, outside the submitted work. Katharine J Looker thanks the National Institute for Health Research Health Protection Research Unit (NIHR HPRU) in Evaluation of Interventions at the University of Bristol, in partnership with Public Health England (PHE), for research support. Katharine J Looker received separate funding from the World Health Organization and Sexual Health 24 during the course of this study. These funders had no role in the writing of the manuscript nor the decision to submit it for publication. The views expressed are those of the authors and not necessarily those of the National Health Service, the NIHR, the Department of Health or Public Health England. Azeem Majid and Imperial College London are grateful for support from the NW London NIHR Collaboration for Leadership in Applied Health Research & Care. Francisco Martins-Melo acknowledges a postdoctoral fellowship from the CAPES (Brazilian Federal Agency for Support and Evaluation of Graduate Education), outside the submitted work. Mohsen Mazidi was supported by the World Academy of Sciences and Chinese Academy of Sciences. John McGrath received John Cade Fellowship APP1056929 from the National Health and Medical Research Council, and Niels Bohr Professorship from the Danish National Research Foundation. Under contract, Ted Miller is leading an independent Measurement and Evaluation of the AB InBev Global Smart Drinking Goals Initiative, which aims to reduce the harmful use of alcohol. The baseline harm levels used in that evaluation will build from GBD's harm estimates. Philip Mitchell's research is supported by an Australian NHMRC Program Grant (no. 1037196). Ulrich Mueller gratefully acknowledges financial Support from the German National Cohort Study (BMBF grant # 01ER1511/D). Olanrewaju Oladimeji is a Senior Research Specialist at the Human Sciences Research Council (HSRC) and Doctoral Candidate at the University of KwaZulu-Natal (UKZN), South Africa; we acknowledge the institutional supports from HSRC and UKZN for him to participate in this study. Alberto Ortiz was supported by the Spanish Government (Intensificacion ISCIIII FEDER funds and RETIC REDINREN RD016/0019). Mayowa Owolabi is supported by U54HG007479 from the NIH. Norberto Perico would like to acknowledge that the work related to this paper has been done on behalf of the GBD Genitourinary Disease Expert Group. Giuseppe Remuzzi acknowledges that the work related to this paper has been done on behalf of the GBD Genitourinary Diseases Expert Group supported by the International Society of Nephrology (ISN). Luz Myriam Reynales-Shigematsu acknowledges the Global Adult Tobacco Survey, GATS Mexico 2015, with financial support provided by the CONADIC, Ministry of Health, Mexico, and the Bloomberg Initiative to Reduce Tobacco Use through the CDC Foundation with a grant from Bloomberg Philanthropies. Aletta E Schutte received support from the South African Medical Research Council and the National Research Foundation's SARChI Programme. Mark Shrime reports grants from the GE Foundation and the the Damon Runyon Cancer Research Foundation GE Safe Surgery 2020 Project. Jasvinder Singh reports consultancy fees from Regeneron, Merz, Iroko, Bioiberica, Crealta/Horizon, Allergan, UBM LLC, WebMD, and the American College of Rheumatology, and consultancy fees and grants from Savient, Takeda. Jasvinder Singh serves as the principal investigator for an investigator-initiated study funded by Horizon pharmaceuticals through a grant to DINORA, Inc, a 501c3 entity; he is also on the steering committee of OMERACT, an international organisation that develops measures for clinical trials and receives arms length funding from 36 pharmaceutical companies. Michael Soljak acknowledges funding from Public Health England for modelling of NCD prevalence. Cassandra Szoeke acknowledges the NHMRC as well as grants from Lundbeck, and Alzheimer's Association, outside the submitted work. In addition, Cassandra Szoeke has a patent, PCT/AU2008/001556 issued. Rafael Tabarés-Seisdedos was supported in part by grant PROMETEOII/2015/021 from Generalitat Valenciana and the national grants PI14/00894 and PIE14/00031 from ISCIII-FEDER. Amand Thrift was supported by a Fellowship from the National Health & Medical Research Council (Australia; 1042600). Stefano Tyrovola's work was supported by the Foundation for Education and European Culture (IPEP), the Sara Borrell postdoctoral programme (reference no. CD15/00019 from the Instituto de Salud Carlos III [ISCIII – Spain]) and the Fondos Europeo de Desarrollo Regional (FEDER). Job van Boven received support from the Department of Clinical Pharmacy and Clinical Pharmacology, University Medical Center Groningen, University of Groningen, Netherlands. Lijing Yan is partially supported by the National Natural Sciences Foundation of China grants (71233001 and 71490732). Marcel Yotebieng is partially supported by the NIAID U01AI096299 and the NICHD R01HD087993.

PY - 2010

Y1 - 2010

N2 - Using the well-known recognition and structural theorem(s) for root-graded Lie algebras and their universal coverings, we give a finite presentation for the universal covering algebra of a centerless Lie torus of type X ≠ A, C, BC. We follow a unified approach for the types under consideration.

AB - Using the well-known recognition and structural theorem(s) for root-graded Lie algebras and their universal coverings, we give a finite presentation for the universal covering algebra of a centerless Lie torus of type X ≠ A, C, BC. We follow a unified approach for the types under consideration.

KW - Extended affine lie algebras

KW - Finite presentation

KW - Lie tori

KW - Root graded Lie algebras

KW - Universal enveloping algebra

UR - http://www.scopus.com/inward/record.url?scp=77956969915&partnerID=8YFLogxK

U2 - 10.2977/PRIMS/16

DO - 10.2977/PRIMS/16

M3 - 学術論文

AN - SCOPUS:77956969915

SN - 0034-5318

VL - 46

SP - 507

EP - 548

JO - Publications of the Research Institute for Mathematical Sciences

JF - Publications of the Research Institute for Mathematical Sciences

IS - 3

ER -

A finite presentation of universal coverings of lie tori

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