A conjecture for the algorithmic decomposition of paths over an SU(3) ADE graph

J.A. Pineda, E. Isasi and M.I. Caicedo

Departamento de Física, Universidad Simón Bolívar, Apartado Postal 89000, Caracas 1080-A, Venezuela.

Received 13 April 2015; ]]> accepted 28 August 2015

Abstract

Through a geometric understanding of the creation, cap, annihilation and cup operators for ADE graphs in SU (3) we propose the first steps towards an algorithm that would allow one to write an arbitrary elementary path as an ordered combination of creation and cap operators acting upon an essential path. We propose a sketch of a proof and use our proposal for some examples for the A₂ and E₅ graphs of the SU (3) family. Attaining this decomposition is an important step in obtaining the path formulation of the quantum Algebra of a modular invariant RCFT.

Keywords: Rational conformal field theory; ADE classification; essential paths; SU (3) Temperley-Lieb algebra; Ocenanu cells; quantum groups; graph theory; integrable systems.

PACS: 02.10.Ox; 02.20.Uw; 02.30.Ik

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