^{1}Laboratoire de Mathématiques, Université Paul Sabatier, Toulouse III, 118 route de Narbonne, 31062 Toulouse, France. e-mail: (email)

^{2}Laboratoire de Mathématiques, Université Blaise Pascal, Clermont-Ferrand II, Campus des Cézeaux, 63177 Aubière Cedex, France. e-mail: (email)

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2009, Issue 626, Pages 75–114, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/CRELLE.2009.003, January 2009

## Abstract

We classify the compact quantum groups acting on 4 points. These are the quantum subgroups of the quantum permutation group 𝓠_{4}. Our main tool is a new presentation for the algebra C(𝓠_{4}), corresponding to an isomorphism of type 𝓠_{4} ≃ *SO*
_{–1}(3). The quantum subgroups of 𝓠_{4} are subject to a McKay type correspondence, that we describe at the level of algebraic invariants.

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