The planar algebra of a fixed point subfactor

Banica, Teodor

Annales mathématiques Blaise Pascal, Tome 25 (2018) no. 2, pp. 247-264.

On considère des inclusions du type ${(P\otimes A)}^G\subset {(P\otimes B)}^G$, où $G$ est un groupe quantique compact de type Kac agissant sur un facteur de type ${\mathrm{II}}_1$ $P$, et sur une inclusion de Markov de $C^{*}$-algèbres de dimension finie $A\subset B$. Dans le cas $[A,B]=0$, qui couvre essentiellement tous les exemples connus, on montre que l’algèbre planaire d’un tel sous-facteur est de la forme $P{(A\subset B)}^G$, avec $G$ agissant dans un certain sens naturel sur l’algèbre de graphe bipartite $P(A\subset B)$.

We consider inclusions of type ${(P\otimes A)}^G\subset {(P\otimes B)}^G$, where $G$ is a compact quantum group of Kac type acting on a ${\mathrm{II}}_1$ factor $P$, and on a Markov inclusion of finite dimensional $C^{*}$-algebras $A\subset B$. In the case $[A,B]=0$, which basically covers all known examples, we show that the planar algebra of such a subfactor is of the form $P{(A\subset B)}^G$, with $G$ acting in some natural sense on the bipartite graph algebra $P(A\subset B)$.

Classification : 46L65, 46L37
Keywords: Compact quantum group, Fixed point subfactor
Mots clés : Groupe quantique compact, Sous-facteur de points fixes

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