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 Hans-Jürgen 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)
Examples of inner linear Hopf algebras.
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.
Balanced bilinear forms and finiteness
properties for incidence coalgebras over a field.
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) quasi-co-Frobenius if and only if C is left (or right) co-Frobenius,
and this is equivalent to the order relation on X being the equality.
Some remarks on Morita theory, Azumaya
algebras and center of an algebra in braided monoidal categories.
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
Representations of finite dimensional pointed
Hopf algebras over S3.
The classification of finite-dimensional pointed Hopf algebras with
group S3 was finished in [AHS]: there are exactly two of them,
the bosonization of a Nichols algebra of dimension 12 and a non-trivial 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 group-likes is a direct sum of
Flatness and freeness properties of the generic
Hopf Galois extensions.
In previous work, to each Hopf algebra H and each invertible
two-cocycle α, Eli Aljadeff and the first-named author attached a subalgebra
of the free commutative Hopf algebra S(tH)Θ
generated by the coalgebra underlying H;
the algebra BαH is the subalgebra of coinvariants of a generic Hopf
Galois extension. In this paper we give conditions under which
is faithfully flat, or even free, as a BαH-module.
We also show that BαH is generated as an algebra by
certain elements arising from the theory of polynomial identities for comodule
algebras developed jointly with Aljadeff.
Tensor categories: a selective guided tour.
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 non-technical, 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 hom-spaces. 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 , which covers less ground in a deeper
Braid representations from quantum groups of
exceptional Lie type.
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.
On the notion of a ribbon quasi-Hopf algebra.
We show that two competing definitions of a ribbon quasi-Hopf
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.
Triangular structures of Hopf algebras and
tensor Morita equivalences.
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 
without polynomial invariants.
Volumen 51, número 2 (2010)
On twisted homogeneous racks of type D.
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 = An,
n ≥ 5, or L a sporadic group.
Global dimensions for Lie groups at level k and
their conformally exceptional quantum subgroups.
We obtain formulae giving global dimensions for fusion categories
defined by Lie groups G at level k and for the associated module-categories
obtained via conformal embeddings. The results can be expressed in terms of
Lie quantum superfactorials of type G which are related, for the type Ar, to
the quantum Barnes function.
Hopf algebras and finite tensor categories in
conformal field theory.
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 finite-dimensional
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.
On semisimple Hopf algebras with few representations
of dimension greater than one.
In the paper we consider semisimple Hopf algebras H with the
following property: irreducible H-modules of the same dimension > 1 are
isomorphic. Suppose that there exists an irreducible H-module M of dimension
> 1 such that its endomorphism ring is a Hopf ideal in H. Then M is the
unique irreducible H-module of dimension > 1.
Drinfel’d doubles and Shapovalov determinants.
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.
Paths on graphs and associated quantum groupoids.
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 A3 and the affine graph A.
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.