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On Frobenius tensor categories Cuadra, Juan


A Hopf algebra H over a field k is called co-Frobenius if it possesses a nonzero right (or left) integral R : H → k. The existence of a nonzero integral amounts to each one of the following conditions on the category C of finite-dimensional right (or left) H-comodules: • C has a nonzero injective object. • C has injective hulls. • C has a nonzero projective object. • C has projective covers. A tensor category satisfying one of these (equivalent) conditions is called Frobenius. In this case, every injective object is projective and vice versa. Radford showed in [3] that a co-Frobenius Hopf algebra whose coradical is a subalgebra has finite coradical filtration. Andruskiewitsch and D˘asc˘alescu proved later in [2] that a Hopf algebra with finite coradical filtration is necessarily co-Frobenius and they conjectured that any co-Frobenius Hopf algebra has finite coradical filtration. In this talk we will show that this conjecture admits a categorical formulation and we will answer it in the affirmative. The idea of the proof is to provide a uniform bound on the composition length of the indecomposable injective objects in terms of the composition series of the injective hull of the unit object. This result is joint with Nicol´as Andruskiewitsch and Pavel Etingof and appears in [1]. References [1] N. Andruskiewitsch, J. Cuadra, and P. Etingof, On two finiteness conditions for Hopf algebras with nonzero integral. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XIV, 401-440. [2] N. Andruskiewitsch and S. D˘asc˘alescu, Co-Frobenius Hopf algebras and the coradical filtration. Math. Z. 243 (2003), 145-154. [3] D. E. Radford, Finiteness conditions for a Hopf algebra with a nonzero integral. J. Algebra 46 (1977), 189-195.

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