The planar algebra of group-type subfactors

Details The planar algebra of group-type subfactors The planar algebra of group-type subfactors

2008-07-25

arxiv.org/abs/0807.4134v2

Mathematics

Operator Algebras

25 pages, 18 figures, To appear in JFA, reviewer's suggestions incorporated

Scientific paper

If $G$ is a countable, discrete group generated by two finite subgroups $H$ and $K$ and $P$ is a II$_1$ factor with an outer G-action, one can construct the group-type subfactor $P^H \subset P \rtimes K$ introduced in \cite{BH}. This construction was used in \cite{BH} to obtain numerous examples of infinite depth subfactors whose standard invariant has exotic growth properties. We compute the planar algebra (in the sense of Jones \cite{J2}) of this subfactor and prove that any subfactor with an abstract planar algebra of "group type" arises from such a subfactor. The action of Jones' planar operad is determined explicitly.

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