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This volume contains the contributed papers from the Colloquium on Hopf algebras
and tensor categories,
which took place from August 31st to September 4th 2009, in La Falda, Sierras de
Córdoba, and the Conference in Honor of Professor HansJürgen Schneider, organized
on the occasion of his 65th birthday, on September 5th 2009 in the Academia
Nacional de Ciencias de Córdoba.
Volumen 51, número 1 (2010)
Preface.
Nicolás Andruskiewitsch and Sonia Natale
PDF

15 
Examples of inner linear Hopf algebras.
Nicolás Andruskiewitsch and Julien Bichon
PDF
The notion of inner linear Hopf algebra is a generalization of the
notion of discrete linear group. In this paper, we prove two general results
that enable us to enlarge the class of Hopf algebras that are known to be inner
linear: the first one is a characterization by using the Hopf dual, while the
second one is a stability result under extensions. We also discuss the related
notion of inner unitary Hopf *algebra.

718 
Balanced bilinear forms and finiteness
properties for incidence coalgebras over a field.
S. Dăscălescu, C. Năstăsescu,
and G. Velicu
PDF
Let C = IC(X) be the incidence coalgebra of an intervally finite
partially ordered set X over a field. We investigate finiteness properties of C.
We determine all C*balanced bilinear forms on C, and we deduce that C is
left (or right) quasicoFrobenius if and only if C is left (or right) coFrobenius,
and this is equivalent to the order relation on X being the equality.

1926 
Some remarks on Morita theory, Azumaya
algebras and center of an algebra in braided monoidal categories.
B. Femić
PDF
We study the left and the right center of an algebra in a braided
monoidal category C. We characterize them by specializing a general result
and point out why in symmetric monoidal categories one does not distinguish
the two centers. We review Morita theorems for a monoidal category D and
analyze necessary conditions for the associativity of the tensor product over an
algebra in D. The central part of the Morita theorems in braided categories is
also discussed. We review the notion of Azumaya algebra in C distinguishing
left and right faithfully projective objects and study the relation between the
two.

2750 
Representations of finite dimensional pointed
Hopf algebras over S_{3}.
Agustín García Iglesias
PDF
The classification of finitedimensional pointed Hopf algebras with
group S_{3} was finished in [AHS]: there are exactly two of them,
the bosonization of a Nichols algebra of dimension 12 and a nontrivial lifting. Here we
determine all simple modules over any of these Hopf algebras. We also find
the Gabriel quivers, the projective covers of the simple modules, and prove
that they are not of finite representation type. To this end, we first investigate
the modules over some complex pointed Hopf algebras defined in the papers
[AG1, GG], whose restriction to the group of grouplikes is a direct sum of
1dimensional modules.

5177 
Flatness and freeness properties of the generic
Hopf Galois extensions.
Christian Kassel and Akira Masuoka
PDF
In previous work, to each Hopf algebra H and each invertible
twococycle α, Eli Aljadeff and the firstnamed author attached a subalgebra
B^{α}_{H}
of the free commutative Hopf algebra S(t_{H})_{Θ}
generated by the coalgebra underlying H;
the algebra B^{α}_{H} is the subalgebra of coinvariants of a generic Hopf
Galois extension. In this paper we give conditions under which
S(t_{H})_{Θ}
is faithfully flat, or even free, as a B^{α}_{H}module.
We also show that B^{α}_{H} is generated as an algebra by
certain elements arising from the theory of polynomial identities for comodule
algebras developed jointly with Aljadeff.

7994 
Tensor categories: a selective guided tour.
Michael Müger
PDF
These are the, somewhat polished and updated, lecture notes for
a three hour course on tensor categories, given at the CIRM, Marseille, in
April 2008. The coverage in these notes is relatively nontechnical, focusing
on the essential ideas. They are meant to be accessible for beginners, but it is
hoped that also some of the experts will find something interesting in them.
Once the basic definitions are given, the focus is mainly on categories that
are linear over a field k and have finite dimensional homspaces. Connections
with quantum groups and low dimensional topology are pointed out, but these
notes have no pretension to cover the latter subjects to any depth. Essentially,
these notes should be considered as annotations to the extensive bibliography.
We also recommend the recent review [43], which covers less ground in a deeper
way.

95163 
Braid representations from quantum groups of
exceptional Lie type.
Eric C. Rowell
PDF
We study the problem of determining if the braid group
representations obtained from quantum groups of types E, F and G at roots of
unity have infinite image or not. In particular we show that when the fusion
categories associated with these quantum groups are not weakly integral, the
braid group images are infinite. This provides further evidence for a recent
conjecture that weak integrality is necessary and sufficient for the braid group
representations associated with any braided fusion category to have finite image.

165175 
On the notion of a ribbon quasiHopf algebra.
Yorck Sommerhäuser
PDF
We show that two competing definitions of a ribbon quasiHopf
algebra are actually equivalent. Along the way, we look at the Drinfel’d element
from a new perspective and use this viewpoint to derive its fundamental properties.

177192 
Triangular structures of Hopf algebras and
tensor Morita equivalences.
Michihisa Wakui
PDF
In this paper, the triangular structures of a Hopf algebra A
are discussed as a tensor Morita invariant. It is shown by many examples that
triangular structures are useful for detecting whether module categories are
monoidally equivalent or not. By counting and comparing the numbers of
triangular structures, we give simple proofs of some results obtained in [25]
without polynomial invariants.

193210 
Volumen 51, número 2 (2010)
On twisted homogeneous racks of type D.
N. Andruskiewitsch, F. Fantino, G. A. García
and L. Vendramin
PDF
We develop some techniques to check when a
twisted homogeneous rack of class (L, t, θ) is of type D.
Then we apply the obtained results to the cases
L = A_{n},
n ≥ 5, or L a sporadic group.

116 
Global dimensions for Lie groups at level k and
their conformally exceptional quantum subgroups.
R. Coquereaux
PDF
We obtain formulae giving global dimensions for fusion categories
defined by Lie groups G at level k and for the associated modulecategories
obtained via conformal embeddings. The results can be expressed in terms of
Lie quantum superfactorials of type G which are related, for the type A_{r}, to
the quantum Barnes function.

1742 
Hopf algebras and finite tensor categories in
conformal field theory.
Jürgen Fuchs and Christoph Schweigert
PDF
In conformal field theory the understanding of
correlation functions can be divided into two distinct conceptual
levels: The analytic properties
of the correlators endow the representation categories of the underlying
chiral symmetry algebras with additional structure, which in suitable cases is
the one of a finite tensor category. The problem of specifying the correlators
can then be encoded in algebraic structure internal to those categories.
After reviewing results for conformal field theories for which these representation
categories are semisimple, we explain what is known about representation
categories of chiral symmetry algebras that are not semisimple. We focus
on generalizations of the Verlinde formula, for which certain finitedimensional
complex Hopf algebras are used as a tool, and on the structural importance
of the presence of a Hopf algebra internal to finite tensor categories.

4390 
On semisimple Hopf algebras with few representations
of dimension greater than one.
V. A. Artamonov
PDF
In the paper we consider semisimple Hopf algebras H with the
following property: irreducible Hmodules of the same dimension > 1 are
isomorphic. Suppose that there exists an irreducible Hmodule M of dimension
> 1 such that its endomorphism ring is a Hopf ideal in H. Then M is the
unique irreducible Hmodule of dimension > 1.

91105 
Drinfel’d doubles and Shapovalov determinants.
I. Heckenberger and H. Yamane
PDF
The Shapovalov determinant for a class of pointed Hopf algebras
is calculated, including quantized enveloping algebras, Lusztig’s small quantum
groups, and quantized Lie superalgebras. Our main tools are root systems,
Weyl groupoids, and Lusztig type isomorphisms. We elaborate powerful
novel techniques for the algebras at roots of unity, and pass to the general case
using a density argument.

107146 
Paths on graphs and associated quantum groupoids.
R. Trinchero
PDF
Given any simple biorientable graph it is shown that there exists a
weak *Hopf algebra constructed on the vector space of graded endomorphisms
of essential paths on the graph. This construction is based on a direct sum
decomposition of the space of paths into orthogonal subspaces one of which is
the space of essential paths. Two simple examples are worked out with certain
detail, the ADE graph A_{3} and the affine graph A_{[2]}.
For the first example the
weak *Hopf algebra coincides with the so called double triangle algebra. No
use is made of Ocneanu’s cell calculus.

147170 
