We complete the realisation by braided subfactors, announced by Ocneanu, of all SU (3)-modular invariant partition functions previously classified by Gannon.

We determine the cells, whose existence has been announced by Ocneanu, on all the candidate nimrep graphs except E (12) 4proposed 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.

We determine spectral measures for some nimrep graphs arising in subfactor theory, particularly those associated with SU (3) modular invariants and subgroups of SU (3). Our methods also give an alternative approach to deriving the results of Banica and Bisch for ADE graphs and subgroups of SU (2) and explain the connection between their results for affine ADE graphs and the Kostant polynomials. We also look at the Hilbert generating series of associated pre-projective algebras.

Abstract. We give a diagrammatic presentation of the A 2 -Temperley-Lieb algebra. Generalizing Jones' notion of a planar algebra, we formulate an A 2 -planar algebra motivated by Kuperberg's A 2 -spider. This A 2 -planar algebra contains a subfamily of vector spaces which will capture the double complex structure pertaining to the subfactor for a finite SU.3/ ADE graph with a flat cell system, including both the periodicity three coming from the A 2 -TemperleyLieb algebra as well as the periodicity two coming from the subfactor basic construction. We use an A 2 -planar algebra to obtain a description of the (Jones) planar algebra for the Wenzl subfactor in terms of generators and relations.Mathematics Subject Classification (2010). Primary 46L37; Secondary 46L60, 81T40.

We determine the Nakayama automorphism of the almost Calabi-Yau algebra A associated to the braided subfactors or nimrep graphs associated to each SU (3) modular invariant. We use this to determine a resolution of A as an A-A bimodule, which will yield a projective resolution of A.