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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todo:paper_03733114_v190_n2_p225_Andruskiewitsch2023-10-03T15:30:18Z Finite-dimensional pointed Hopf algebras with alternating groups are trivial Andruskiewitsch, N. Fantino, F. Graña, M. Vendramin, L. 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_03733114_v190_n2_p225_Andruskiewitsch |
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Universidad de Buenos Aires |
institution_str |
I-28 |
repository_str |
R-134 |
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
topic |
Nichols algebras Pointed Hopf algebras Racks |
spellingShingle |
Nichols algebras Pointed Hopf algebras Racks Andruskiewitsch, N. Fantino, F. Graña, M. Vendramin, L. 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. |
format |
JOUR |
author |
Andruskiewitsch, N. Fantino, F. Graña, M. Vendramin, L. |
author_facet |
Andruskiewitsch, N. Fantino, F. Graña, M. Vendramin, L. |
author_sort |
Andruskiewitsch, N. |
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 |
url |
http://hdl.handle.net/20.500.12110/paper_03733114_v190_n2_p225_Andruskiewitsch |
work_keys_str_mv |
AT andruskiewitschn finitedimensionalpointedhopfalgebraswithalternatinggroupsaretrivial AT fantinof finitedimensionalpointedhopfalgebraswithalternatinggroupsaretrivial AT granam finitedimensionalpointedhopfalgebraswithalternatinggroupsaretrivial AT vendraminl finitedimensionalpointedhopfalgebraswithalternatinggroupsaretrivial |
_version_ |
1782029843413598208 |