Larson-Sweedler Theorem and the Role of Grouplike Elements in Weak Hopf Algebras

We extend the Larson-Sweedler theorem to 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 show that the category of modules over a weak Hopf algebra is autonomous monoidal with semisimple unit and invertible modules. We also reveal the connection of invertible modules to left and right grouplike elements in the dual weak Hopf algebra. Defining distinguished left and right grouplike elements we derive the Radford formula for the fourth power of the antipode in a weak Hopf algebra and prove that the order of the antipode is finite up to an inner automorphism by a grouplike element in the trivial subalgebra A^T of the underlying weak Hopf algebra A.Comment: version appeared in J.Algebra, 45 pages, plain TeX, extended introduction, shortened proof

S_4-symmetry of 6j-symbols and Frobenius-Schur indicators in rigid monoidal C^*-categories

We show that a left-rigid monoidal C^*-category with irreducible monoidal unit is also a sovereign and spherical category. Defining a Frobenius-Schur type indicator we obtain selection rules for the fusion coefficients of irreducible objects. As a main result we prove S_4-invariance of 6j-symbols in such a category.Comment: 21 pages + 16 pages with figures; LaTeX2e plus macro package XYpic; file with included pictures available as http://www.desy.de/~jfuchs/s4/s4.ps.g

Addressing, Greeting and Related Gestures in the Opening Sequences of Finnish, French and Hungarian YouTube Videos

This paper compares the opening sequences of Finnish, French and Hungarian YouTube videos. We concentrate on addressing, greeting and related gestures, important elements when YouTubers interact with their imagined viewers, using data consisting of 138 videos in the three languages. This study falls within the field of the pragmatics of social media and interpersonal pragmatics, and data were analysed using multimodal discourse analysis. Shared practices included the frequent use of greetings, a preference for general nominal address forms as well as for iconic and deictic gestures. Cross-cultural differences revealed that Finnish and Hungarian were closer to each other than to French. Shared practices may be connected to the genre of YouTube videos, whereas differences appear related to cross-cultural practices generally.Peer reviewe

Flow limitation and riverbank protection design using asymmetrical flow mapping on a physical hydraulic model

The experimental monitoring of an asymmetrical flow pattern, realised on a physical model at the Laboratory for Applied Hydraulics of HEPIA, yielded accurate dimensioning of a flow limitation device and appropriate riverbank protection design. The studied structures were then implemented on the Aire River in Geneva. The main goals of the Aire River revitalization program in Geneva are: hazard and risk mitigation. Inundation risk is mitigated for the Q300y=120 m3/s design discharge by an orifice-weir structure yielding a 400'000 m3 flood retention devices [1] preserving the orifice of driftwood clogging. Since upstream from the orifice flow conditions are strongyle asymmetrical, velocity field needed to be monitored on a physical hydraulic model [2]. Velocity measurement was carried out by means of Met-Flow UVP probes and the shear stress calculated. The experimental analysis results yielded an appropriate orifice geometry and riverbank protection design

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 theories. In this paper the essential structure (hidden by a large gauge freedom) of rational Hopf algebras is revealed. This allows one to construct examples of rational Hopf algebras starting only from the corresponding fusion ring. In particular we classify all solutions for fusion rules with not more than three sectors, as well as for the level 3 affine A$^{(1)}_1$ fusion rules