On finite dimensional Nichols algebras of diagonal type

Fil: Andruskiewitsch, Nicolás. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.

Detalles Bibliográficos
Autores principales: Andruskiewitsch, Nicolás, Angiono, Iván Ezequiel
Otros Autores: https://orcid.org/0000-0002-9163-5161
Formato: publishedVersion article
Lenguaje:Inglés
Publicado: 2024
Materias:
Nichols algebras
Quantum groups
Weyl groupoid
Modular Lie algebras
Acceso en línea:http://hdl.handle.net/11086/553004
https://doi.org/10.1007/s13373-017-0113-x
Aporte de:
Repositorio Digital Universitario (UNC) de Universidad Nacional de Córdoba
id I10-R141-11086-553004
record_format dspace
institution Universidad Nacional de Córdoba
institution_str I-10
repository_str R-141
collection Repositorio Digital Universitario (UNC)
language Inglés
topic Nichols algebras
Quantum groups
Weyl groupoid
Modular Lie algebras
spellingShingle Nichols algebras
Quantum groups
Weyl groupoid
Modular Lie algebras
Andruskiewitsch, Nicolás
Angiono, Iván Ezequiel
On finite dimensional Nichols algebras of diagonal type
topic_facet Nichols algebras
Quantum groups
Weyl groupoid
Modular Lie algebras
description Fil: Andruskiewitsch, Nicolás. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.
author2 https://orcid.org/0000-0002-9163-5161
author_facet https://orcid.org/0000-0002-9163-5161
Andruskiewitsch, Nicolás
Angiono, Iván Ezequiel
format publishedVersion
article
author Andruskiewitsch, Nicolás
Angiono, Iván Ezequiel
author_sort Andruskiewitsch, Nicolás
title On finite dimensional Nichols algebras of diagonal type
title_short On finite dimensional Nichols algebras of diagonal type
title_full On finite dimensional Nichols algebras of diagonal type
title_fullStr On finite dimensional Nichols algebras of diagonal type
title_full_unstemmed On finite dimensional Nichols algebras of diagonal type
title_sort on finite dimensional nichols algebras of diagonal type
publishDate 2024
url http://hdl.handle.net/11086/553004
https://doi.org/10.1007/s13373-017-0113-x
work_keys_str_mv AT andruskiewitschnicolas onfinitedimensionalnicholsalgebrasofdiagonaltype
AT angionoivanezequiel onfinitedimensionalnicholsalgebrasofdiagonaltype
_version_ 1806949050663043072
spelling I10-R141-11086-5530042024-08-01T06:34:44Z On finite dimensional Nichols algebras of diagonal type Andruskiewitsch, Nicolás Angiono, Iván Ezequiel https://orcid.org/0000-0002-9163-5161 https://orcid.org/0000-0001-8767-1648 Nichols algebras Quantum groups Weyl groupoid Modular Lie algebras info:eu-repo/semantics/publishedVersion Fil: Andruskiewitsch, Nicolás. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina. Fil: Andruskiewitsch, Nicolás. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Fil: Andruskiewitsch, Nicolás. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina. Fil: Angiono, Iván Ezequiel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina. Fil: Angiono, Iván Ezequiel. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Fil: Angiono, Iván Ezequiel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina. This is a survey on Nichols algebras of diagonal type with finite dimension, or more generally with arithmetic root system. The knowledge of these algebras is the cornerstone of the classification program of pointed Hopf algebras with finite dimension, or finite Gelfand–Kirillov dimension; and their structure should be indispensable for the understanding of the representation theory, the computation of the various cohomologies, and many other aspects of finite dimensional pointed Hopf algebras. These Nichols algebras were classified in Heckenberger (Adv Math 220:59–124, 2009) as a notable application of the notions of Weyl groupoid and generalized root system (Heckenberger in Invent Math 164:175–188, 2006; Heckenberger and Yamane in Math Z 259:255–276, 2008). In the first part of this monograph, we give an overview of the theory of Nichols algebras of diagonal type. This includes a discussion of the notion of generalized root system and its appearance in the contexts of Nichols algebras of diagonal type and (modular) Lie superalgebras. In the second and third part, we describe for each Nichols algebra in the list of Heckenberger (2009) the following basic information: the generalized root system; its label in terms of Lie theory; the defining relations found in Angiono (J Eur Math Soc 17:2643–2671, 2015; J Reine Angew Math 683:189–251, 2013); the PBW-basis; the dimension or the Gelfand–Kirillov dimension; the associated Lie algebra as in Andruskiewitsch et al. (Bull Belg Math Soc Simon Stevin 24(1):15–34, 2017). Indeed the second part deals with Nichols algebras related to Lie algebras and superalgebras in arbitrary characteristic, while the third contains the information on Nichols algebras related to Lie algebras and superalgebras only in small characteristic, and the few examples yet unidentified in terms of Lie theory. The work of Nicolás Andruskiewitsch and Iván Angiono was partially supported by CONICET, Secyt (UNC), the MathAmSud Project GR2HOPF. The work of Iván Angiono was partially supported by ANPCyT (Foncyt). The work of Nicolás Andruskiewitsch, respectively Iván Angiono, was partially done during a visit to the University of Hamburg, respectively the MPI (Bonn), supported by the Alexander von Humboldt Foundation. info:eu-repo/semantics/publishedVersion Fil: Andruskiewitsch, Nicolás. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina. Fil: Andruskiewitsch, Nicolás. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Fil: Andruskiewitsch, Nicolás. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina. Fil: Angiono, Iván Ezequiel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina. Fil: Angiono, Iván Ezequiel. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Fil: Angiono, Iván Ezequiel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina. Matemática Pura 2024-07-31T12:37:21Z 2024-07-31T12:37:21Z 2017 article Andruskiewitsch, N. y Angiono, I. E. (2017). On finite dimensional Nichols algebras of diagonal type. Bulletin of Mathematical Sciences, 7, 353-573. https://doi.org/10.1007/s13373-017-0113-x 1664-3607 http://hdl.handle.net/11086/553004 1664-3615 https://doi.org/10.1007/s13373-017-0113-x eng Attribution 4.0 International http://creativecommons.org/licenses/by/4.0/ Impreso; Electrónico y/o Digital

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