A2-Planar Algebras II: Planar Modules

Generalizing Jones’s notion of a planar algebra, we have previously introduced an A_2-planar algebra capturing the structure contained in the double complex pertaining to the subfactor for a finite SU(3) ADE graph with a flat cell system. We now introduce the notion of modules over an A_2-planar algebra, and describe certain irreducible Hilbert A_2-TL-modules. We construct an A_2-graph planar algebra associated to each pair (G,W) given by an SU(3) ADE graph G and a cell system W on G. A partial modular decomposition of these A_2-graph planar algebras is achieved.