2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/196374We use Cvitanovic's diagrammatic techniques to construct the rational solutions of the Yang-Baxter equation associated with the $e_6$ and $e_7$ families of Lie algebras, and thus explain Westbury's observations about their uniform spectral decompositions. In doing so we explore the extensions of the Brauer and symmetric group algebras to the centralizer algebras of $e_7$ and $e_6$ on their lowest-dimensional representations and (up to three-fold) tensor products thereof, giving bases for them and a number of identities satisfied by the algebras' defining invariant tensors.24 pages. More minor correctionsQuantum AlgebraHigh Energy Physics - TheoryRepresentation Theorytext