Paths on graphs and associated quantum groupoids

TL;DR

The paper constructs weak {*}-Hopf algebras from simple biorientable graphs using essential paths, providing explicit examples and connecting to known algebraic structures without relying on Ocneanu's cell calculus.

Contribution

It introduces a new method to build weak {*}-Hopf algebras from graphs based on path decompositions, expanding the algebraic understanding of graph-related quantum structures.

Findings

Constructed weak {*}-Hopf algebras on graph path spaces.

Explicit examples with ADE $A_3$ and affine $A_{[2]}$ graphs.

Identified the algebra for $A_3$ with the double triangle algebra.

Abstract

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.