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.

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.

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.

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 1-dimensional modules.

In previous work, to each Hopf algebra H and each invertible two-cocycle α, Eli Aljadeff and the first-named 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.

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 [43], which covers less ground in a deeper way.

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.

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.

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.

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.

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 A_r, to the quantum Barnes function.

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.

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.

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.

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_[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.