Finite-dimensional pointed Hopf algebras with alternating groups are trivial

It is shown that Nichols algebras over alternating groups Am (m ≥ 5) are infinite dimensional. This proves that any complex finite dimensional pointed Hopf algebra with group of group-likes isomorphic to Am is isomorphic to the group algebra. In a similar fashion, it is shown that the Nichols algebr...

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Autores principales:	Andruskiewitsch, Nicolás, Graña, Matías Alejo, Vendramin, Leandro
Publicado:	2011
Materias:	Nichols algebras Pointed Hopf algebras Racks
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03733114_v190_n2_p225_Andruskiewitsch http://hdl.handle.net/20.500.12110/paper_03733114_v190_n2_p225_Andruskiewitsch
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_03733114_v190_n2_p225_Andruskiewitsch
record_format	dspace
spelling	paper:paper_03733114_v190_n2_p225_Andruskiewitsch2023-06-08T15:37:55Z Finite-dimensional pointed Hopf algebras with alternating groups are trivial Andruskiewitsch, Nicolás Graña, Matías Alejo Vendramin, Leandro Nichols algebras Pointed Hopf algebras Racks It is shown that Nichols algebras over alternating groups Am (m ≥ 5) are infinite dimensional. This proves that any complex finite dimensional pointed Hopf algebra with group of group-likes isomorphic to Am is isomorphic to the group algebra. In a similar fashion, it is shown that the Nichols algebras over the symmetric groups Sm are all infinite-dimensional, except maybe those related to the transpositions considered in Fomin and Kirillov (Progr Math 172:146-182, 1999), and the class of type (2, 3) in S5. We also show that any simple rack X arising from a symmetric group, with the exception of a small list, collapse, in the sense that the Nichols algebra B(X, q) is infinite dimensional, q an arbitrary cocycle. © 2010 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag. Fil:Andruskiewitsch, N. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Graña, M. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Vendramin, L. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2011 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03733114_v190_n2_p225_Andruskiewitsch http://hdl.handle.net/20.500.12110/paper_03733114_v190_n2_p225_Andruskiewitsch
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Nichols algebras Pointed Hopf algebras Racks
spellingShingle	Nichols algebras Pointed Hopf algebras Racks Andruskiewitsch, Nicolás Graña, Matías Alejo Vendramin, Leandro Finite-dimensional pointed Hopf algebras with alternating groups are trivial
topic_facet	Nichols algebras Pointed Hopf algebras Racks
description	It is shown that Nichols algebras over alternating groups Am (m ≥ 5) are infinite dimensional. This proves that any complex finite dimensional pointed Hopf algebra with group of group-likes isomorphic to Am is isomorphic to the group algebra. In a similar fashion, it is shown that the Nichols algebras over the symmetric groups Sm are all infinite-dimensional, except maybe those related to the transpositions considered in Fomin and Kirillov (Progr Math 172:146-182, 1999), and the class of type (2, 3) in S5. We also show that any simple rack X arising from a symmetric group, with the exception of a small list, collapse, in the sense that the Nichols algebra B(X, q) is infinite dimensional, q an arbitrary cocycle. © 2010 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag.
author	Andruskiewitsch, Nicolás Graña, Matías Alejo Vendramin, Leandro
author_facet	Andruskiewitsch, Nicolás Graña, Matías Alejo Vendramin, Leandro
author_sort	Andruskiewitsch, Nicolás
title	Finite-dimensional pointed Hopf algebras with alternating groups are trivial
title_short	Finite-dimensional pointed Hopf algebras with alternating groups are trivial
title_full	Finite-dimensional pointed Hopf algebras with alternating groups are trivial
title_fullStr	Finite-dimensional pointed Hopf algebras with alternating groups are trivial
title_full_unstemmed	Finite-dimensional pointed Hopf algebras with alternating groups are trivial
title_sort	finite-dimensional pointed hopf algebras with alternating groups are trivial
publishDate	2011
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03733114_v190_n2_p225_Andruskiewitsch http://hdl.handle.net/20.500.12110/paper_03733114_v190_n2_p225_Andruskiewitsch
work_keys_str_mv	AT andruskiewitschnicolas finitedimensionalpointedhopfalgebraswithalternatinggroupsaretrivial AT granamatiasalejo finitedimensionalpointedhopfalgebraswithalternatinggroupsaretrivial AT vendraminleandro finitedimensionalpointedhopfalgebraswithalternatinggroupsaretrivial
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Finite-dimensional pointed Hopf algebras with alternating groups are trivial

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