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On Frobenius tensor categories Cuadra, Juan
Description
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.
Item Metadata
| Title |
On Frobenius tensor categories
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| Creator | |
| Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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| Date Issued |
2016-08-18T16:30
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| Description |
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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| Extent |
34 minutes
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| Subject | |
| Type | |
| File Format |
video/mp4
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| Language |
eng
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| Notes |
Author affiliation: University of Almeria
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| Series | |
| Date Available |
2017-02-17
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| Provider |
Vancouver : University of British Columbia Library
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| Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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| DOI |
10.14288/1.0342811
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| URI | |
| Affiliation | |
| Peer Review Status |
Unreviewed
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| Scholarly Level |
Faculty
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| Rights URI | |
| Aggregated Source Repository |
DSpace
|
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International