Equivalence of categories of simple Lie algebras in positive characteristic
In this paper we first study some properties of the finite-dimensional simple restricted Lie algebras. In the literature is found a one-to-one correspondence between them and finite-dimensional simple Lie algebras over a field of positive characteristic. Motivated by this fact, we give a one-to-one...
- Autores:
-
Guevara, Carlos Rafael
Quintero Vanegas, Elkin O.
Benítez Monsalve, Germán Alfonso
- Tipo de recurso:
- Fecha de publicación:
- 2022
- Institución:
- Universidad Tecnológica de Bolívar
- Repositorio:
- Repositorio Institucional UTB
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.utb.edu.co:20.500.12585/12440
- Acceso en línea:
- https://hdl.handle.net/20.500.12585/12440
- Palabra clave:
- Restricted Lie algebras,
Restricted Simple Lie algebras,
Simple restricted Lie algebras,
Equivalence of categories
LEMB
- Rights
- openAccess
- License
- http://creativecommons.org/publicdomain/zero/1.0/
| id |
UTB2_13e819fd4cfed2dd19d57c4b7d737027 |
|---|---|
| oai_identifier_str |
oai:repositorio.utb.edu.co:20.500.12585/12440 |
| network_acronym_str |
UTB2 |
| network_name_str |
Repositorio Institucional UTB |
| repository_id_str |
|
| dc.title.spa.fl_str_mv |
Equivalence of categories of simple Lie algebras in positive characteristic |
| title |
Equivalence of categories of simple Lie algebras in positive characteristic |
| spellingShingle |
Equivalence of categories of simple Lie algebras in positive characteristic Restricted Lie algebras, Restricted Simple Lie algebras, Simple restricted Lie algebras, Equivalence of categories LEMB |
| title_short |
Equivalence of categories of simple Lie algebras in positive characteristic |
| title_full |
Equivalence of categories of simple Lie algebras in positive characteristic |
| title_fullStr |
Equivalence of categories of simple Lie algebras in positive characteristic |
| title_full_unstemmed |
Equivalence of categories of simple Lie algebras in positive characteristic |
| title_sort |
Equivalence of categories of simple Lie algebras in positive characteristic |
| dc.creator.fl_str_mv |
Guevara, Carlos Rafael Quintero Vanegas, Elkin O. Benítez Monsalve, Germán Alfonso |
| dc.contributor.author.none.fl_str_mv |
Guevara, Carlos Rafael Quintero Vanegas, Elkin O. Benítez Monsalve, Germán Alfonso |
| dc.subject.keywords.spa.fl_str_mv |
Restricted Lie algebras, Restricted Simple Lie algebras, Simple restricted Lie algebras, Equivalence of categories |
| topic |
Restricted Lie algebras, Restricted Simple Lie algebras, Simple restricted Lie algebras, Equivalence of categories LEMB |
| dc.subject.armarc.none.fl_str_mv |
LEMB |
| description |
In this paper we first study some properties of the finite-dimensional simple restricted Lie algebras. In the literature is found a one-to-one correspondence between them and finite-dimensional simple Lie algebras over a field of positive characteristic. Motivated by this fact, we give a one-to-one correspondence between their morphisms, which allow us to conclude that such categories are equivalent |
| publishDate |
2022 |
| dc.date.issued.none.fl_str_mv |
2022-07-18 |
| dc.date.accessioned.none.fl_str_mv |
2023-07-27T19:43:18Z |
| dc.date.available.none.fl_str_mv |
2023-07-27T19:43:18Z |
| dc.date.submitted.none.fl_str_mv |
2023-07-27 |
| dc.type.coarversion.fl_str_mv |
http://purl.org/coar/version/c_970fb48d4fbd8a85 |
| dc.type.driver.spa.fl_str_mv |
info:eu-repo/semantics/article |
| dc.type.hasversion.spa.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
| dc.type.spa.spa.fl_str_mv |
http://purl.org/coar/resource_type/c_2df8fbb1 |
| status_str |
publishedVersion |
| dc.identifier.citation.spa.fl_str_mv |
Monsalve, G. B., Guevara, C. R. P., & Vanegas, E. Q. (2023). Equivalence of categories of simple Lie algebras in positive characteristic. Proyecciones (Antofagasta, On line), 42(4), 815-831. |
| dc.identifier.uri.none.fl_str_mv |
https://hdl.handle.net/20.500.12585/12440 |
| dc.identifier.doi.none.fl_str_mv |
10.22199/issn.0717-6279-5335 |
| dc.identifier.instname.spa.fl_str_mv |
Universidad Tecnológica de Bolívar |
| dc.identifier.reponame.spa.fl_str_mv |
Repositorio Universidad Tecnológica de Bolívar |
| identifier_str_mv |
Monsalve, G. B., Guevara, C. R. P., & Vanegas, E. Q. (2023). Equivalence of categories of simple Lie algebras in positive characteristic. Proyecciones (Antofagasta, On line), 42(4), 815-831. 10.22199/issn.0717-6279-5335 Universidad Tecnológica de Bolívar Repositorio Universidad Tecnológica de Bolívar |
| url |
https://hdl.handle.net/20.500.12585/12440 |
| dc.language.iso.spa.fl_str_mv |
eng |
| language |
eng |
| dc.rights.coar.fl_str_mv |
http://purl.org/coar/access_right/c_abf2 |
| dc.rights.uri.*.fl_str_mv |
http://creativecommons.org/publicdomain/zero/1.0/ |
| dc.rights.accessrights.spa.fl_str_mv |
info:eu-repo/semantics/openAccess |
| dc.rights.cc.*.fl_str_mv |
CC0 1.0 Universal |
| rights_invalid_str_mv |
http://creativecommons.org/publicdomain/zero/1.0/ CC0 1.0 Universal http://purl.org/coar/access_right/c_abf2 |
| eu_rights_str_mv |
openAccess |
| dc.format.extent.none.fl_str_mv |
17 páginas |
| dc.format.mimetype.spa.fl_str_mv |
application/pdf |
| dc.coverage.spatial.none.fl_str_mv |
2022-2023, COLOMBIA, CARTAGENA DE INDIAS |
| dc.publisher.place.spa.fl_str_mv |
Cartagena de Indias |
| dc.publisher.sede.spa.fl_str_mv |
Campus Tecnológico |
| dc.source.spa.fl_str_mv |
Proyecciones Journal of Mathematics |
| institution |
Universidad Tecnológica de Bolívar |
| bitstream.url.fl_str_mv |
https://repositorio.utb.edu.co/bitstream/20.500.12585/12440/1/5335-Article%20Text-34632-1-10-20230707.pdf https://repositorio.utb.edu.co/bitstream/20.500.12585/12440/2/license_rdf https://repositorio.utb.edu.co/bitstream/20.500.12585/12440/3/license.txt https://repositorio.utb.edu.co/bitstream/20.500.12585/12440/4/5335-Article%20Text-34632-1-10-20230707.pdf.txt https://repositorio.utb.edu.co/bitstream/20.500.12585/12440/5/5335-Article%20Text-34632-1-10-20230707.pdf.jpg |
| bitstream.checksum.fl_str_mv |
b6cbb8e6bc7d62502839cb9d0c015cf6 42fd4ad1e89814f5e4a476b409eb708c e20ad307a1c5f3f25af9304a7a7c86b6 560d68c8b6222c11de81cbf47a7360fb 3d1d672ff3915bb3cfb0d024edf8da66 |
| bitstream.checksumAlgorithm.fl_str_mv |
MD5 MD5 MD5 MD5 MD5 |
| repository.name.fl_str_mv |
Repositorio Institucional UTB |
| repository.mail.fl_str_mv |
repositorioutb@utb.edu.co |
| _version_ |
1814021632228851712 |
| spelling |
Guevara, Carlos Rafael40819dbb-a7d9-45dd-b58f-616d3d3e13c1Quintero Vanegas, Elkin O.6001ab13-21fb-4a7b-9f99-acef57a7757bBenítez Monsalve, Germán Alfonso9c1fd30f-d3f9-4a63-bcc8-d366b9b02d8c2022-2023, COLOMBIA, CARTAGENA DE INDIAS2023-07-27T19:43:18Z2023-07-27T19:43:18Z2022-07-182023-07-27Monsalve, G. B., Guevara, C. R. P., & Vanegas, E. Q. (2023). Equivalence of categories of simple Lie algebras in positive characteristic. Proyecciones (Antofagasta, On line), 42(4), 815-831.https://hdl.handle.net/20.500.12585/1244010.22199/issn.0717-6279-5335Universidad Tecnológica de BolívarRepositorio Universidad Tecnológica de BolívarIn this paper we first study some properties of the finite-dimensional simple restricted Lie algebras. In the literature is found a one-to-one correspondence between them and finite-dimensional simple Lie algebras over a field of positive characteristic. Motivated by this fact, we give a one-to-one correspondence between their morphisms, which allow us to conclude that such categories are equivalent17 páginasapplication/pdfenghttp://creativecommons.org/publicdomain/zero/1.0/info:eu-repo/semantics/openAccessCC0 1.0 Universalhttp://purl.org/coar/access_right/c_abf2Proyecciones Journal of MathematicsEquivalence of categories of simple Lie algebras in positive characteristicinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_2df8fbb1http://purl.org/coar/version/c_970fb48d4fbd8a85Restricted Lie algebras,Restricted Simple Lie algebras,Simple restricted Lie algebras,Equivalence of categoriesLEMBCartagena de IndiasCampus TecnológicoPúblico generalR. E. Block. The classification problem for simple Lie algebras of characteristic p. In: D. Winter, D. (eds) Lie Algebras and Related Topics. Lecture Notes in Mathematics, 933. Berlin: Springer, 1982. doi: 10.1007/BFb0093351R. E. Block, and R. L. Wilson, “The restricted simple Lie algebras are of classical or Cartan type”, Proceedings of the National Academy of Sciences, vol. 81, no. 16, pp. 5271-5274, 1984. doi: 10.1073/pnas.81.16.5271R. E. Block, and R. L. Wilson, “Classification of the restricted simple Lie algebras”, Journal of Algebra, vol. 114, no. 1, pp. 115-259, 1988. doi: 10.1016/0021-8693(88)90216-5N. Jacobson, Lie algebras. Interscience Tracts in Pure and Applied Mathematics, no. 10 Interscience New York: John Wiley & Sons, Inc., 1962V. G. Kac, “Description of the filtered Lie algebras with which graded Lie algebras of Cartan type are associated”, Mathematics of the USSR-Izvestiya, vol. 38, pp. 800-834, 1974. doi: 10.1070/IM1974v008n04ABEH002128A. Premet, and H. Strade, “Simple Lie algebras of small characteristic. I. Sandwich elements”, Journal of Algebra, vol. 189, no. 2, 419-480, 1997. doi: 10.1006/jabr.1996.6861A. Premet, and H. Strade, “Simple Lie algebras of small characteristic. II. Exceptional roots”, Journal of Algebra, vol. 216, no. 1, pp. 190-301, 1999. doi: 10.1006/jabr.1998.7746A. Premet, and H. Strade, “Simple Lie algebras of small characteristic. III. The toral rank 2 case”, Journal of Algebra, vol. 242, no. 1, pp. 236-337, 2001. doi: 10.1006/jabr.2001.8806A. Premet, and H. Strade, “Simple Lie algebras of small characteristic. IV. Solvable and classical roots”, Journal of Algebra, vol. 278, no. 2, pp. 766-833, 2004. doi: 10.1016/j.jalgebra.2003.10.028A. Premet, and H. Strade, Classification of finite dimensional simple Lie algebras in prime characteristics. In: Representations of algebraicgroups, quantum groups, and Lie algebras. Contemporary Mathematics, vol. 413. Providence, RI: American Mathematical Society, 2006A. Premet, and H. Strade, “Simple Lie algebras of small characteristic. V. The non-Melikian case”, Journal of Algebra, vol. 314, no. 2, pp. 664-692, 2007. doi: 10.1016/j.jalgebra.2007.02.059A. Premet, and H. Strade, “Simple Lie algebras of small characteristic. VI. Completion of the classification”, Journal of Algebra, vol. 320, no. 9, pp. 3559-3604, 2008. doi: 10.1016/j.jalgebra.2008.08.012S. Skryabin, “Toral rank one simple Lie algebras of low characteristics”, Journal of Algebra, vol. 200, no. 2, pp. 650-700, 1998. doi: 10.1006/jabr.1997.7231H. Strade, “The classification of the simple modular Lie algebras. III. Solution of the classical case”, Annals of Mathematics, vol. 133, no. 3, pp. 577-604, 1991. doi: 10.2307/2944320H. Strade, “The classification of the simple modular Lie algebras. II. The toral structure”, Journal of Algebra, vol. 151, no. 2, pp. 425-475, 1992. doi: 10.1016/0021-8693(92)90122-3H. Strade, Simple Lie algebras over fields of positive characteristic. vol. I: Structure theory. De Gruyter Expositions in Mathematics, 38. Walter de Gruyter & Co., Berlin, 2004H. Strade, and R. Farnsteiner, Modular Lie algebras and their representations. Monographs and Textbooks in Pure and Applied Mathematics, 116. Marcel Dekker, Inc., New York, 1988H. Strade, and R. L. Wilson, “Classification of simple Lie algebras over algebraically closed fields of prime characteristic”, Bulletin of the American Mathematical Society, vol. 24, no. 2, pp. 357-362, 1991.F. Viviani, “Infinitesimal deformations of restricted simple Lie algebras. I”, Journal of Algebra, vol. 320, no. 12, pp. 4102-4131, 2008. doi: 10.1016/j.jalgebra.2008.08.022http://purl.org/coar/resource_type/c_2df8fbb1ORIGINAL5335-Article Text-34632-1-10-20230707.pdf5335-Article Text-34632-1-10-20230707.pdfArtículo principalapplication/pdf270177https://repositorio.utb.edu.co/bitstream/20.500.12585/12440/1/5335-Article%20Text-34632-1-10-20230707.pdfb6cbb8e6bc7d62502839cb9d0c015cf6MD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8701https://repositorio.utb.edu.co/bitstream/20.500.12585/12440/2/license_rdf42fd4ad1e89814f5e4a476b409eb708cMD52LICENSElicense.txtlicense.txttext/plain; charset=utf-83182https://repositorio.utb.edu.co/bitstream/20.500.12585/12440/3/license.txte20ad307a1c5f3f25af9304a7a7c86b6MD53TEXT5335-Article Text-34632-1-10-20230707.pdf.txt5335-Article Text-34632-1-10-20230707.pdf.txtExtracted texttext/plain27296https://repositorio.utb.edu.co/bitstream/20.500.12585/12440/4/5335-Article%20Text-34632-1-10-20230707.pdf.txt560d68c8b6222c11de81cbf47a7360fbMD54THUMBNAIL5335-Article Text-34632-1-10-20230707.pdf.jpg5335-Article Text-34632-1-10-20230707.pdf.jpgGenerated Thumbnailimage/jpeg3638https://repositorio.utb.edu.co/bitstream/20.500.12585/12440/5/5335-Article%20Text-34632-1-10-20230707.pdf.jpg3d1d672ff3915bb3cfb0d024edf8da66MD5520.500.12585/12440oai:repositorio.utb.edu.co:20.500.12585/124402023-07-28 00:17:32.812Repositorio Institucional UTBrepositorioutb@utb.edu.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 |