Poisson boundaries  of monoidal categories

Neshveyev, Sergey; Yamashita, Makoto

doi:10.24033/asens.2335

Poisson boundaries of monoidal categories
[Les frontières de Poisson des catégories monoïdales]

Neshveyev, Sergey ; Yamashita, Makoto

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 4, pp. 927-972.

Résumé
Abstract

Etant données une C $^{*}$ -catégorie tensorielle rigide $𝒞$ dont l'objet unité est simple ainsi qu'une mesure de probabilité $μ$ sur l'ensemble de classes d'isomorphisme des objets simples, nous définissons la frontière de Poisson de $(𝒞, μ)$ . C'est une nouvelle C $^{*}$ -catégorie tensorielle $𝒫$ dont l'objet unité n'est pas, en général, simple, couplée avec un foncteur unitaire tensoriel $Π : 𝒞 \to 𝒫$ . Notre résultat principal assure que si l'objet unité de $𝒫$ est simple (ce qui se traduit par une condition sur une certaine marche aléatoire classique), alors $Π$ est un foncteur unitaire tensoriel universel qui définit la fonction de dimension moyennable sur $𝒞$ . Les corollaires de ce théorème unifient différents résultats connus sur la moyennabilité des C $^{*}$ -catégories tensorielles, des groupes quantiques et des sous-facteurs.

Given a rigid C $^{*}$ -tensor category $𝒞$ with simple unit and a probability measure $μ$ on the set of isomorphism classes of its simple objects, we define the Poisson boundary of $(𝒞, μ)$ . This is a new C $^{*}$ -tensor category $𝒫$ , generally with nonsimple unit, together with a unitary tensor functor $Π : 𝒞 \to 𝒫$ . Our main result is that if $𝒫$ has simple unit (which is a condition on some classical random walk), then $Π$ is a universal unitary tensor functor defining the amenable dimension function on $𝒞$ . Corollaries of this theorem unify various results in the literature on amenability of C $^{*}$ -tensor categories, quantum groups, and subfactors.

MR Zbl

DOI : 10.24033/asens.2335

Classification : 18D10; 60J50, 46L50.
Mots-clés : Monoidal category, random walk, Poisson boundary, catégorie monoïdale, marche aléatoire, frontière de Poisson.

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     title = {Poisson boundaries  of monoidal categories},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {927--972},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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Neshveyev, Sergey; Yamashita, Makoto. Poisson boundaries  of monoidal categories. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 4, pp. 927-972. doi : 10.24033/asens.2335. http://archive.numdam.org/articles/10.24033/asens.2335/

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