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**1 - 5**of**5**### Cardiff University,

, 2009

"... We determine the cells, whose existence has been announced by Ocneanu, on all the candidate nimrep graphs except E (12) 4 proposed by di Francesco and Zuber for the SU(3) modular invariants classified by Gannon. This enables the Boltzmann weights to be computed for the corresponding integrable stati ..."

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We determine the cells, whose existence has been announced by Ocneanu, on all the candidate nimrep graphs except E (12) 4 proposed by di Francesco and Zuber for the SU(3) modular invariants classified by Gannon. This enables the Boltzmann weights to be computed for the corresponding integrable statistical mechanical models and provide the framework for studying corresponding braided subfactors to realise all the SU(3) modular invariants as well as a framework for a new SU(3) planar algebra theory. 1

### School of Mathematics, Cardiff University,

, 2009

"... Generalizing Jones’s notion of a planar algebra, we have previously introduced an A2-planar algebra capturing the structure contained in the SU(3) ADE subfactors. We now introduce the notion of modules over an A2-planar algebra, and describe certain irreducible Hilbert A2-TL-modules. A partial decom ..."

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Generalizing Jones’s notion of a planar algebra, we have previously introduced an A2-planar algebra capturing the structure contained in the SU(3) ADE subfactors. We now introduce the notion of modules over an A2-planar algebra, and describe certain irreducible Hilbert A2-TL-modules. A partial decomposition of the A2planar algebras for the ADE nimrep graphs associated to SU(3) modular invariants is achieved. 1