Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4
After the classification of simple Lie algebras over a field of characteristic p > 3, the main problem not yet solved in the theory of finite dimensional Lie algebras is the classification of simple Lie algebras over a field of characteristic 2. The first result for this classification problem en...
- Autores:
-
Payares Guevara, Carlos R.
Arias Amaya, Fabián
- Tipo de recurso:
- Fecha de publicación:
- 2021
- Institución:
- Universidad Tecnológica de Bolívar
- Repositorio:
- Repositorio Institucional UTB
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.utb.edu.co:20.500.12585/10366
- Acceso en línea:
- https://hdl.handle.net/20.500.12585/10366
- Palabra clave:
- Simple Lie 2-algebra
Toral rank
Classical type lie algebra
Contragredient lie algebra
LEMB
- Rights
- openAccess
- License
- http://creativecommons.org/licenses/by-nc-nd/4.0/
| id |
UTB2_c1e601423ba83bc22a0045220ffd6b47 |
|---|---|
| oai_identifier_str |
oai:repositorio.utb.edu.co:20.500.12585/10366 |
| network_acronym_str |
UTB2 |
| network_name_str |
Repositorio Institucional UTB |
| repository_id_str |
|
| dc.title.spa.fl_str_mv |
Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4 |
| title |
Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4 |
| spellingShingle |
Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4 Simple Lie 2-algebra Toral rank Classical type lie algebra Contragredient lie algebra LEMB |
| title_short |
Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4 |
| title_full |
Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4 |
| title_fullStr |
Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4 |
| title_full_unstemmed |
Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4 |
| title_sort |
Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4 |
| dc.creator.fl_str_mv |
Payares Guevara, Carlos R. Arias Amaya, Fabián |
| dc.contributor.author.none.fl_str_mv |
Payares Guevara, Carlos R. Arias Amaya, Fabián |
| dc.subject.keywords.spa.fl_str_mv |
Simple Lie 2-algebra Toral rank Classical type lie algebra Contragredient lie algebra |
| topic |
Simple Lie 2-algebra Toral rank Classical type lie algebra Contragredient lie algebra LEMB |
| dc.subject.armarc.none.fl_str_mv |
LEMB |
| description |
After the classification of simple Lie algebras over a field of characteristic p > 3, the main problem not yet solved in the theory of finite dimensional Lie algebras is the classification of simple Lie algebras over a field of characteristic 2. The first result for this classification problem ensures that all finite dimensional Lie algebras of absolute toral rank 1 over an algebraically closed field of characteristic 2 are soluble. Describing simple Lie algebras (respectively, Lie 2-algebras) of finite dimension of absolute toral rank (respectively, toral rank) 3 over an algebraically closed field of characteristic 2 is still an open problem. In this paper we show that there are no classical type simple Lie 2-algebras with toral rank odd and furthermore that the simple contragredient Lie 2-algebra G(F4,a) of dimension 34 has toral rank 4. Additionally, we give the Cartan decomposition of G(F4,a). |
| publishDate |
2021 |
| dc.date.accessioned.none.fl_str_mv |
2021-09-22T21:27:13Z |
| dc.date.available.none.fl_str_mv |
2021-09-22T21:27:13Z |
| dc.date.issued.none.fl_str_mv |
2021-04-29 |
| dc.date.submitted.none.fl_str_mv |
2021-09-08 |
| dc.type.driver.spa.fl_str_mv |
info:eu-repo/semantics/article |
| dc.type.hasversion.spa.fl_str_mv |
info:eu-repo/semantics/restrictedAccess |
| dc.type.spa.spa.fl_str_mv |
http://purl.org/coar/resource_type/c_2df8fbb1 |
| dc.identifier.citation.spa.fl_str_mv |
Payares Guevara, Carlos R. y Fabián A. Arias Amaya. "Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4" Revista de La Unión Matemática Argentina , vol. 62, no. 1, 29 de abril de 2021, págs. 123-139, https://doi.org/10.33044/revuma.1555. |
| dc.identifier.uri.none.fl_str_mv |
https://hdl.handle.net/20.500.12585/10366 |
| dc.identifier.doi.none.fl_str_mv |
10.33044/revuma.1555 |
| 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 |
Payares Guevara, Carlos R. y Fabián A. Arias Amaya. "Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4" Revista de La Unión Matemática Argentina , vol. 62, no. 1, 29 de abril de 2021, págs. 123-139, https://doi.org/10.33044/revuma.1555. 10.33044/revuma.1555 Universidad Tecnológica de Bolívar Repositorio Universidad Tecnológica de Bolívar |
| url |
https://hdl.handle.net/20.500.12585/10366 |
| 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/licenses/by-nc-nd/4.0/ |
| dc.rights.accessrights.spa.fl_str_mv |
info:eu-repo/semantics/openAccess |
| dc.rights.cc.*.fl_str_mv |
Attribution-NonCommercial-NoDerivatives 4.0 Internacional |
| rights_invalid_str_mv |
http://creativecommons.org/licenses/by-nc-nd/4.0/ Attribution-NonCommercial-NoDerivatives 4.0 Internacional 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 |
Argentina |
| dc.publisher.place.spa.fl_str_mv |
Cartagena de Indias |
| dc.source.spa.fl_str_mv |
Revista de la Unión Matemática Argentina, Vol. 62, No. 1, 2021 |
| institution |
Universidad Tecnológica de Bolívar |
| bitstream.url.fl_str_mv |
https://repositorio.utb.edu.co/bitstream/20.500.12585/10366/1/CLASSICAL%20SIMPLE%20LIE%202-ALGEBRAS%20OF%20ODD%20TORAL_Carlos%20Rafael%20Payare.pdf https://repositorio.utb.edu.co/bitstream/20.500.12585/10366/2/license_rdf https://repositorio.utb.edu.co/bitstream/20.500.12585/10366/3/license.txt https://repositorio.utb.edu.co/bitstream/20.500.12585/10366/4/CLASSICAL%20SIMPLE%20LIE%202-ALGEBRAS%20OF%20ODD%20TORAL_Carlos%20Rafael%20Payare.pdf.txt https://repositorio.utb.edu.co/bitstream/20.500.12585/10366/5/CLASSICAL%20SIMPLE%20LIE%202-ALGEBRAS%20OF%20ODD%20TORAL_Carlos%20Rafael%20Payare.pdf.jpg |
| bitstream.checksum.fl_str_mv |
fd1ac7dcd4ad254d732e96710682cbb4 4460e5956bc1d1639be9ae6146a50347 e20ad307a1c5f3f25af9304a7a7c86b6 05c9aef16edc7144820d9187ea40d25a 8fbaa5cbefc420b1e16eaa6d0cb2f496 |
| 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_ |
1814021705294675968 |
| spelling |
Payares Guevara, Carlos R.ba2913ef-00b7-4541-9329-dc960d04b2cfArias Amaya, Fabián713ecd4c-d974-4280-a4cd-5129cdc3e781Argentina2021-09-22T21:27:13Z2021-09-22T21:27:13Z2021-04-292021-09-08Payares Guevara, Carlos R. y Fabián A. Arias Amaya. "Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4" Revista de La Unión Matemática Argentina , vol. 62, no. 1, 29 de abril de 2021, págs. 123-139, https://doi.org/10.33044/revuma.1555.https://hdl.handle.net/20.500.12585/1036610.33044/revuma.1555Universidad Tecnológica de BolívarRepositorio Universidad Tecnológica de BolívarAfter the classification of simple Lie algebras over a field of characteristic p > 3, the main problem not yet solved in the theory of finite dimensional Lie algebras is the classification of simple Lie algebras over a field of characteristic 2. The first result for this classification problem ensures that all finite dimensional Lie algebras of absolute toral rank 1 over an algebraically closed field of characteristic 2 are soluble. Describing simple Lie algebras (respectively, Lie 2-algebras) of finite dimension of absolute toral rank (respectively, toral rank) 3 over an algebraically closed field of characteristic 2 is still an open problem. In this paper we show that there are no classical type simple Lie 2-algebras with toral rank odd and furthermore that the simple contragredient Lie 2-algebra G(F4,a) of dimension 34 has toral rank 4. Additionally, we give the Cartan decomposition of G(F4,a).17 páginasapplication/pdfenghttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessAttribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://purl.org/coar/access_right/c_abf2Revista de la Unión Matemática Argentina, Vol. 62, No. 1, 2021Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4info:eu-repo/semantics/articleinfo:eu-repo/semantics/restrictedAccesshttp://purl.org/coar/resource_type/c_2df8fbb1Simple Lie 2-algebraToral rankClassical type lie algebraContragredient lie algebraLEMBCartagena de IndiasA. Grishkov and A. Premet, Simple Lie algebras of absolute toral rank 2 in characteristic 2, Preprint. https://www.ime.usp.br/˜grishkov/papers/asp.pdf.A. Grishkov, On simple Lie algebras over a field of characteristic 2, J. Algebra 363 (2012), 14–18. MR 2925843.S. P. Demuˇskin, Cartan subalgebras of the simple Lie p-algebras Wn and Sn, Sibirsk. Mat. Z. ˇ 11 (1970), 310–325. MR 0262310G. M. D. Hogeweij, Almost-classical Lie algebras. I, II, Nederl. Akad. Wetensch. Indag. Math. 44 (1982), no. 4, 441–452, 453–460. MR 0683531.N. Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers, New York, 1962. MR 0143793.N. Jacobson, Abstract derivation and Lie algebras, Trans. Amer. Math. Soc. 42 (1937), no. 2, 206–224. MR 1501922V. G. Kac, The classification of the simple Lie algebras over a field with non-zero characteristic, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 385–408. MR 0276286I. Kaplansky, Linear algebra and geometry. A second course, Allyn and Bacon, Boston, MA, 1969. MR 0249444.G. B. Seligman, On Lie algebras of prime characteristic, Mem. Amer. Math. Soc. 19 (1956). MR 0077876.S. Skryabin, Toral rank one simple Lie algebras of low characteristics, J. Algebra 200 (1998), no. 2, 650–700. MR 1610680R. Steinberg, Automorphisms of classical Lie algebras, Pacific J. Math. 11 (1961), 1119–1129. MR 0143845H. Strade, The absolute toral rank of a Lie algebra, in Lie algebras, Madison 1987, 1–28, Lecture Notes in Math., 1373, Springer, Berlin, 1989. MR 1007321.B. Ju. Ve˘ısfe˘ıler and V. G. Kac, Exponentials in Lie algebras of characteristic p, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 762–788. MR 0306282http://purl.org/coar/resource_type/c_2df8fbb1ORIGINALCLASSICAL SIMPLE LIE 2-ALGEBRAS OF ODD TORAL_Carlos Rafael Payare.pdfCLASSICAL SIMPLE LIE 2-ALGEBRAS OF ODD TORAL_Carlos Rafael Payare.pdfapplication/pdf436295https://repositorio.utb.edu.co/bitstream/20.500.12585/10366/1/CLASSICAL%20SIMPLE%20LIE%202-ALGEBRAS%20OF%20ODD%20TORAL_Carlos%20Rafael%20Payare.pdffd1ac7dcd4ad254d732e96710682cbb4MD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8805https://repositorio.utb.edu.co/bitstream/20.500.12585/10366/2/license_rdf4460e5956bc1d1639be9ae6146a50347MD52LICENSElicense.txtlicense.txttext/plain; charset=utf-83182https://repositorio.utb.edu.co/bitstream/20.500.12585/10366/3/license.txte20ad307a1c5f3f25af9304a7a7c86b6MD53TEXTCLASSICAL SIMPLE LIE 2-ALGEBRAS OF ODD TORAL_Carlos Rafael Payare.pdf.txtCLASSICAL SIMPLE LIE 2-ALGEBRAS OF ODD TORAL_Carlos Rafael Payare.pdf.txtExtracted texttext/plain36597https://repositorio.utb.edu.co/bitstream/20.500.12585/10366/4/CLASSICAL%20SIMPLE%20LIE%202-ALGEBRAS%20OF%20ODD%20TORAL_Carlos%20Rafael%20Payare.pdf.txt05c9aef16edc7144820d9187ea40d25aMD54THUMBNAILCLASSICAL SIMPLE LIE 2-ALGEBRAS OF ODD TORAL_Carlos Rafael Payare.pdf.jpgCLASSICAL SIMPLE LIE 2-ALGEBRAS OF ODD TORAL_Carlos Rafael Payare.pdf.jpgGenerated Thumbnailimage/jpeg60189https://repositorio.utb.edu.co/bitstream/20.500.12585/10366/5/CLASSICAL%20SIMPLE%20LIE%202-ALGEBRAS%20OF%20ODD%20TORAL_Carlos%20Rafael%20Payare.pdf.jpg8fbaa5cbefc420b1e16eaa6d0cb2f496MD5520.500.12585/10366oai:repositorio.utb.edu.co:20.500.12585/103662023-05-25 11:41:58.977Repositorio Institucional UTBrepositorioutb@utb.edu.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 |