Quantum automorphism groups of small metric spaces

To any ?nite metric space X we associate the universal Hopf C*-algebra H coacting on X. We prove that spaces X having at most 7 points fall into one of the following classes: (1) the coaction of H is not transitive; (2) H is the algebra of functions on the automorphism group of X; (3) X is a simplex and H corresponds to a Temperley?Lieb algebra; (4) X is a product of simplices and H corresponds to a Fuss?Catalan algebra.

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