Planar depth and planar subalgebras (Q1865294)

In [Planar algebras. I (preprint math.QA/9909027)] planar algebras were introduced by \textit{V. F. R. Jones} to get insight in the study of subfactors, especially in the algebraic-combinatorial aspects of the lattice of higher relative commutants. To each extremal type \(II_1\) finite index subfactor inclusion, Jones has associated a spherical \(C^*\)-planar algebra structure. In the paper under review the authors first define a natural notion of planar depth for planar algebras. Then they give a sufficient condition for the finiteness of this depth which is easy to check on the principal graph of the planar algebra. Let \(G\) be a group which acts outerly on a \(II_1\) factor \(R\). The planar algebra \(P(G)\) of the corresponding fixed-points subfactor was previously studied by the first author in [\textit{Z. A. Landau}, Geom. Dedicata 95, 183-214 (2002; Zbl 1022.46039)]. Here, the case of an additional automorphic action of a finite group \(\Theta\) on \(G\) is taken into account. It is proved that the planar subalgebra \(P^\Theta\) of \(P(G)\), defined by the invariants of the action of \(\Theta\) on \(P(G)\), is isomorphic to the planar algebra \(P^{N\subset M}\), where \(N=R^{G\propto \Theta} \subset M=R^\Theta\). Finally the planar depth of \(P^\Theta\) is discussed in a number of examples.