Quantum groups based on spatial partitions

Cébron, Guillaume; Weber, Moritz

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 32 (2023) no. 4, pp. 727-768

We define new compact matrix quantum groups whose intertwiner spaces are dual to tensor categories of three-dimensional set partitions (which we call spatial partitions). This extends substantially Banica and Speicher’s approach of the so called easy quantum groups: It enables us to find new examples of quantum subgroups of Wang’s free orthogonal quantum group $O_n^{+}$ which do not contain the symmetric group $S_n$; we may define new kinds of products of quantum groups coming from new products of categories of partitions; and we give a quantum group interpretation of certain categories of partitions which do neither contain the pair partition nor the identity partition.

Nous définissons de nouveaux groupes quantiques compacts de matrices dont les espaces d’entrelaceurs sont en dualité avec des catégories tensorielles de partitions d’ensembles tri-dimensionels (que nous appelons partitions spatiales). Cela généralise de manière conséquente l’approche de Banica et Speicher dite des groupes quantiques « easy » : cela nous permet d’exhiber de nouveaux exemples de sous-groupes quantiques du groupe quantique orthogonal libre $O_n^{+}$ de Wang qui ne contiennent pas le groupe symétrique $S_n$ ; nous pouvons définir de nouveaux types de produits de groupes quantiques, venant de nouveaux produits de catégories de partitions ; et nous donnons une interprétation en terme de groupe quantique de certaines catégories de partitions qui ne contiennent ni la partition paire, ni la partition identité.

Classification : 20G42, 05A18, 05E10, 46L54
Keywords: set partitions, three-dimensional partitions, spatial partitions, compact matrix quantum groups, easy quantum groups, partition quantum groups, Banica–Speicher quantum groups, free orthogonal quantum groups, tensor categories, Kronecker product

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