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

Selective melanocortin MC4 receptor agonists reverse haemorrhagic shock and prevent multiple organ damage

Background and purpose: In circulatory shock, melanocortins have life-saving effects likely to be mediated by MC4 receptors. To gain direct insight into the role of melanocortin MC4 receptors in haemorrhagic shock, we investigated the effects of two novel selective MC4 receptor agonists. Experimental approach: Severe haemorrhagic shock was produced in rats under general anaesthesia. Rats were then treated with either the non-selective agonist [Nle(4), D-Phe(7)]alpha-melanocyte-stimulating hormone (NDP-alpha-MSH) or with the selective MC4 agonists RO27-3225 and PG-931. Cardiovascular and respiratory functions were continuously monitored for 2 h; survival rate was recorded up to 24 h. Free radicals in blood were measured using electron spin resonance spectrometry; tissue damage was evaluated histologically 25 min or 24 h after treatment. Key results: All shocked rats treated with saline died within 30-35 min. Treatment with NDP-alpha-MSH, RO27-3225 and PG-931 produced a dose-dependent (13-108 nmol kg(-1) i.v.) restoration of cardiovascular and respiratory functions, and improved survival. The three melanocortin agonists also markedly reduced circulating free radicals relative to saline-treated shocked rats. All these effects were prevented by i.p. pretreatment with the selective MC4 receptor antagonist HS024. Moreover, treatment with RO27-3225 prevented morphological and immunocytochemical changes in heart, lung, liver, and kidney, at both early (25 min) and late (24 h) intervals. Conclusions and Implications: Stimulation of MC4 receptors reversed haemorrhagic shock, reduced multiple organ damage and improved survival. Our findings suggest that selective MC4 receptor agonists could have a protective role against multiple organ failure following circulatory shock