%0 Generic %1 Trinchero2010 %A Trinchero, R. %D 2010 %K graphs groupoids paths quantum %T Paths on graphs and associated quantum groupoids* %U http://arxiv.org/abs/1004.5104 %X Given any simple biorientable graph it is shown that there exists a weak *-Hopf algebra constructed out of the space of paths on the corresponding 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.

@misc{Trinchero2010, abstract = { Given any simple biorientable graph it is shown that there exists a weak {*}-Hopf algebra constructed out of the space of paths on the corresponding 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. }, added-at = {2010-04-29T12:29:53.000+0200}, author = {Trinchero, R.}, biburl = {https://www.bibsonomy.org/bibtex/2e7dc37e383bcd077c1c70e3159f693ec/uludag}, description = {Paths on graphs and associated quantum groupoids*}, interhash = {f5ae8926d557204c76ab6e69d2a2dd7e}, intrahash = {e7dc37e383bcd077c1c70e3159f693ec}, keywords = {graphs groupoids paths quantum}, note = {cite arxiv:1004.5104 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.}, timestamp = {2010-04-29T12:29:53.000+0200}, title = {Paths on graphs and associated quantum groupoids*}, url = {http://arxiv.org/abs/1004.5104}, year = 2010 }