Finite-dimensional pointed Hopf algebras with alternating groups are trivial

By N. Andruskiewitsch, F. Fantino, M. Graña, Leandro Vendramin. 2008

74 citations recorded by the source corpus.

It is shown that Nichols algebras over alternating groups $${\mathbb A_m}$$ (m ≥ 5) are infinite dimensional. This proves that any complex finite dimensional pointed Hopf algebra with group of group-likes isomorphic to $${\mathbb A_m}$$ is isomorphic to the group algebra. In a similar fashion, it is shown that the Nichols algebras over the symmetric groups $${\mathbb S_m}$$ 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 $${\mathbb S_5}$$. 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 $${\mathfrak B(X, \bf q)}$$ is infinite dimensional, q an arbitrary cocycle.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c70d289c926973bc7a7e01f3f741f3213ae8cc49

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Citations in the corpus

On finite-dimensional Hopf algebras (2014, background): “An indecomposable rack X collapses when dimB(X,q) = ∞ for every finite faithful 373 2-cocycle q (see [7] for the definition of faithful).”
Hopf Algebras and Root Systems (2020, methodology): “The rack theoretical formulation of Corol-lary 17.1.11 (ﬁnite racks of type D collapse) was used for example in [ AF + 11a ] to show that any ﬁnite-dimensional pointed Hopf algebra H over C with group G ( H ) ∼ = A n , n ≥ 5, is isomorphic to the group algebra CA n of the alternating group.”