## From paths on graphs to quantum groupoids I

### Robert Coquereaux – 30 August 2006

We shall present and discuss the notion of essential paths on graphs (Ocneanu) and use this notion to define the (first) algebra structure on the quantum groupoid associated with a graph G (here, we shall take G to be a tree). This algebra is finite dimenional when G is of type ADE. We shall also discuss the relations between essential paths, representation theory of SU2, Temperlie - Lieb - Jones algebras and with the Gelfand - Ponomarev construction for quivers. Genralizations to SLn will also be discussed.

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