2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61439Given a planar algebra we show the equivalence of the notions of a module over this algebra (in the operadic sense), and module over a universal annular algebra. We classify such modules, with invariant inner products, in the generic region and give applications to subfactorss, including a planar construction of the $E_6$ and $E_8$ subfactors.62 pages, includes 2 tables and 24 figures, to be published in L'Enseignement MathematiqueOperator AlgebrasQuantum Algebra46L37; 57M25text