Co-Frobenius Hopf algebras and the coradical filtration

Andruskiewitsch, Nicolás; Dăscălescu, S.

doi:10.1007/s00209-002-0456-0

Cited by 33 publications

(19 citation statements)

References 24 publications

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“…Theorem 2.5. The following assertions are equivalent: (iv) ⇔ (v) The proof of this equivalence given in [AD,page 148] when H 0 is a subalgebra is also valid in this setting. The argument is as follows.…”

Section: Co-frobenius Hopf Algebrasmentioning

confidence: 84%

“…(3) Let G be a connected, simply connected, simple complex algebraic group and let ǫ be a primitive ℓ-th root of 1, ℓ odd and 3 ∤ ℓ if G is of type G 2 . It was shown in [AD,Example 4.1] using the Hopf socle that the quantum group O ǫ (G) is co-Frobenius. An alternative proof follows from Theorem 2.10.…”

Section: (I) If H Is Co-frobenius and H Is Injective As Right K-comodmentioning

confidence: 99%

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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: Co-frobenius Hopf Algebrasmentioning

confidence: 84%

Section: (I) If H Is Co-frobenius and H Is Injective As Right K-comodmentioning

confidence: 99%

On the structure of (co-Frobenius) Hopf algebras

Andruskiewitsch¹,

Cuadra²

2013

J. Noncommut. Geom.

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“…In [4], an alternative proof to Radford's theorem was given, together with an analysis of the structure of a co-Frobenius Hopf algebra whose coradical is a Hopf subalgebra, along the lines of the method proposed in [5]; see also [6]. However, there are examples of co-Frobenius Hopf algebras whose coradical is not a Hopf subalgebra.…”

Section: Introductionmentioning

confidence: 99%

“…A prominent one is the function algebra O q (G) of a semisimple quantum group G at a root of one q; it was shown in [2] that the injective hulls of the simple comodules are finite dimensional. Another approach appears in [4] through the notion of Hopf socle. Assume that H has bijective antipode.…”

Section: Introductionmentioning

confidence: 99%

On two finiteness conditions for Hopf algebras with nonzero integral

Andruskiewitsch¹,

Cuadra²,

Etingof³

2015

Annali Scuola Normale Superiore - Classe Di Scienze

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A Hopf algebra is co-Frobenius when it has a nonzero integral. It is proved that the composition length of the indecomposable injective comodules over a co-Frobenius Hopf algebra is bounded. As a consequence, the coradical filtration of a co-Frobenius Hopf algebra is finite; this confirms a conjecture by Sorin Dȃscȃlescu and the first author. The proof is of categorical nature and the same result is obtained for Frobenius tensor categories of subexponential growth. A family of co-Frobenius Hopf algebras that are not of finite type over their Hopf socles is constructed, answering so in the negative another question by the same authors.

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“…For a Hopf algebra H , it is known that H being co-Frobenius is equivalent to H having a non-zero right integral, so this property defines this class of Hopf algebras. Hopf algebras with non-zero integrals are very important and have been intensely studied in the last years, mainly because they have very good structural and representation theoretic properties (see [AD,DNR,H1,S1,S2,Sw1,Sw2] and references therein). Quantum groups with non-zero integrals are also of great interest; see [APW,AD,H1,H2,H3].…”

mentioning

confidence: 99%

Co-Frobenius coalgebras

Iovanov

2006

Journal of Algebra

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We investigate left and right co-Frobenius coalgebras and give equivalent characterizations which prove statements dual to the characterizations of Frobenius algebras. We prove that a coalgebra is left and right coFrobenius if and only if C ∼ = Rat(C * C * ) as right C * -modules and also that this is equivalent to the fact that the functors Hom K (−, K) and Hom C * (−, C * ) from M C to C * M are isomorphic. This allows a definition of a left-right symmetric concept of co-Frobenius coalgebras that is perfectly dual to the one of Frobenius algebras and coincides to the existing notion left and right co-Frobenius coalgebra.

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Co-Frobenius Hopf algebras and the coradical filtration

Abstract: We prove that a Hopf algebra with a finite coradical filtration is co-Frobenius. We also characterize co-Frobenius Hopf algebras with corad-

Cited by 33 publications

References 24 publications

On the structure of (co-Frobenius) Hopf algebras

On the structure of (co-Frobenius) Hopf algebras

On two finiteness conditions for Hopf algebras with nonzero integral

Co-Frobenius coalgebras

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