A2-Planar Algebras II: Planar Modules

TL;DR

This paper extends the theory of A_2-planar algebras by introducing modules, constructing associated graph planar algebras, and analyzing their modular decomposition, advancing the understanding of algebraic structures related to SU(3) ADE graphs.

Contribution

It introduces modules over A_2-planar algebras, constructs A_2-graph planar algebras for SU(3) ADE graphs, and explores their modular decomposition, providing new tools for algebraic and subfactor theory.

Findings

Defined irreducible Hilbert A_2-TL-modules.

Constructed A_2-graph planar algebras for each (G,W).

Achieved partial modular decomposition of these algebras.

Abstract

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.