On two finiteness conditions for Hopf algebras with nonzero integral

Andruskiewitsch, Nicolás; Cuadra, Juan; Etingof, Pavel

doi:10.48550/arxiv.1206.5934

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2013

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“…Related to this problem, in Theorem 2.5 we also show that a Hopf algebra is co-Frobenius if and only if the Hopf coradical is so and its standard filtration is finite. After acceptance of the present paper, it was proved in [ACE,Theorem 1.2] that the conjecture is true.…”

Section: Introductionmentioning

confidence: 66%

On the structure of (co-Frobenius) Hopf algebras

Andruskiewitsch¹,

Cuadra²

2013

J. Noncommut. Geom.

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Abstract. We introduce a new filtration on Hopf algebras, the standard filtration, generalizing the coradical filtration. Its zeroth term, called the Hopf coradical, is the subalgebra generated by the coradical. We give a structure theorem: any Hopf algebra with injective antipode is a deformation of the bosonization of the Hopf coradical by its diagram, a connected graded Hopf algebra in the category of Yetter-Drinfeld modules over the latter. We discuss the steps needed to classify Hopf algebras in suitable classes accordingly. For the class of co-Frobenius Hopf algebras, we prove that a Hopf algebra is co-Frobenius if and only if its Hopf coradical is so and the diagram is finite dimensional. We also prove that the standard filtration of such Hopf algebras is finite. Finally, we show that extensions of co-Frobenius (resp. cosemisimple) Hopf algebras are co-Frobenius (resp. cosemisimple).

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Section: Introductionmentioning

confidence: 66%