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Paths on graphs and associated quantum groupoids*

Apr, 2010
21 pages
Published in:
  • Rev.Union Mat.Argentina 51 (2010) 2
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Abstract: (arXiv)
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 A3A_{3} and the affine graph A[2]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.
Note:
  • 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 included
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