Weak Hopf Algebras: I. Integral Theory and C-Structure

@article{Bhm1998WeakHA,
  title={Weak Hopf Algebras: I. Integral Theory and C-Structure},
  author={Gabriella B{\`o}hm and Florian Nill and Korn{\'e}l Szlach{\'a}nyi},
  journal={Journal of Algebra},
  year={1998},
  volume={221},
  pages={385-438}
}
We give an introduction to the theory of weak Hopf algebras proposed as a coassociative alternative of weak quasi-Hopf algebras. We follow an axiomatic approach keeping as close as possible to the “classical” theory of Hopf algebras. The emphasis is put on the new structure related to the presence of canonical subalgebras AL and AR in any weak Hopf algebra A that play the role of non-commutative numbers in many respects. A theory of integrals is developed in which we show how the algebraic… 

ON THE REPRESENTATIONS OF WEAK CROSSED PRODUCTS

Let H be a finite-dimensional weak Hopf algebra in the sense of [G. Bohm and K. Szlachanyi, A coassociative C*-quantum group with nonintegral dimensions, Lett. Math. Phys.35 (1996) 437–456] which is

Semisimple Hopf algebras and their depth two Hopf subalgebras

We prove that a depth two Hopf subalgebra K of a semisimple Hopf algebra H is normal (where the ground field $k$ is algebraically closed of characteristic zero). This means on the one hand that a

On Weak Crossed Products of Weak Hopf Algebras

Let H be a weak Hopf algebra in the sense of Böhm et al. (J Algebra 221:385–438, 1999) measuring an algebra A. Let A#σH be a weak crossed product with σ invertible. Then in this paper we first give

Larson-Sweedler Theorem, Grouplike Elements, Invertible Modules and the Order of the Antipode in Weak Hopf Algebras

We extend the Larson–Sweedler theorem for weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a non-degenerate left integral. We establish

Weak Hopf Algebras: II. Representation Theory, Dimensions, and the Markov Trace

Abstract If A is a weak C *-Hopf algebra then the category of finite-dimensional unitary representations of A is a monoidal C *-category with its monoidal unit being the GNS representation D e

Hopf algebroids with balancing subalgebra

The Morita contexts and galois extensions for weak Hopf group coalgebras

In this paper, we mainly construct a Morita context associated to a weak Hopf group coalgebra and as application we also compute the Morita contexts of weak Hopf group Galois extensions. 1. The first

The formal theory of Hopf algebras Part II: The case of Hopf algebras

Abstract The category HopfR of Hopf algebras over a commutative unital ring R is analyzed with respect to its categorical properties. The main results are: (1) For every ring R the category HopfR is
...

References

SHOWING 1-10 OF 41 REFERENCES

Weak Hopf Algebras: II. Representation Theory, Dimensions, and the Markov Trace

Abstract If A is a weak C *-Hopf algebra then the category of finite-dimensional unitary representations of A is a monoidal C *-category with its monoidal unit being the GNS representation D e

Weak Hopf Algebras and Reducible Jones Inclusions of Depth 2. I: From Crossed Products to Jones Towers

We apply the theory of finite dimensional weak C � -Hopf algebras A as developed by G. Bohm, F. Nill and K. Szlachanyi (BSz,Sz,N1,N2,BNS) to study reducible inclusion triples of von-Neumann algebras

Weak (Hopf) Bialgebras

In this chapter, the R-bialgebroids and Hopf algebroids of the previous chapter are investigated further in the particular case when the base algebra R carries a so-called separable Frobenius

Operator Algebras and Applications: Quantized groups, string algebras, and Galois theory for algebras

We introduce a Galois type invariant for the position of a subalgebra inside an algebra, called a paragroup, which has a group-like structure. Paragroups are the natural quantization of (finite)

The Haar measure on finite quantum groups

By a finite quantum group, we will mean in this paper a finitedimensional Hopf algebra. A left Haar measure on such a quantum group is a linear functional satisfying a certain invariance property. In

Quasi Hopf quantum symmetry in quantum theory

RATIONAL HOPF ALGEBRAS: POLYNOMIAL EQUATIONS, GAUGE FIXING, AND LOW-DIMENSIONAL EXAMPLES

Rational Hopf algebras, i.e. certain quasitriangular weak quasi-Hopf algebras whose representations form a tortile modular C* category, are expected to describe the quantum symmetry of rational field

Compact matrix pseudogroups

The compact matrix pseudogroup is a non-commutative compact space endowed with a group structure. The precise definition is given and a number of examples is presented. Among them we have compact

Quantum group symmetry of partition functions of IRF models and its application to Jones' index theory

For each Boltzmann weight of a face model, we associate two quantum groups (face algebras) which describe the dependence of the partition function on boundary value condition. Using these, we give a