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 $A_{3}$ 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.Comment: To appear in the proceedings of "Colloquium on Hopf Algebras, Quantum Groups and Tensor Categories", August 31st to September 4th 2009, La Falda, Cordoba, Argentina. Additional clarifying remarks has been include

Critical behavior of a nonlocal φ4 field theory and asymptotic freedom

The critical behavior of a nonlocal scalar field theory is studied. This theory has a nonlocal kinetic term which involves a real power 1-2α of the Laplacian. The interaction term is the usual local φ4 interaction. The lowest order Feynman diagrams corresponding to coupling constant renormalization, mass renormalization, and field renormalization are computed. Particular features appearing in the renormalization of this nonlocal theory that differ from the case of local theories are studied. The previous calculations lead to the perturbative computation of the coupling constant beta function and critical exponents ν and η. In four dimensions for α<0 this beta function presents asymptotic freedom in the UV. This is remarkable since no non-Abelian vector fields are included. However, this comes at the expense of losing reflection positivity.Fil: Trinchero, Roberto Carlos. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; Argentin

On quantum symmetries of ADE graphs

The double triangle algebra(DTA) associated to an ADE graph is considered. A description of its bialgebra structure based on a reconstruction approach is given. This approach takes as initial data the representation theory of the DTA as given by Ocneanu's cell calculus. It is also proved that the resulting DTA has the structure of a weak *-Hopf algebra. As an illustrative example, the case of the graph A3 is described in detail.Comment: 15 page