Rational R-matrices, centralizer algebras and tensor identities for e6 and e7 exceptional families of Lie algebras

Abstract

We use Cvitanović'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 range of identities satisfied by the algebras' defining invariant tensors. 1 nm15@york.ac.uk

Group Theory

“…Also in 2002, an intriguing link between the q-state Potts models and the E 8 family was discovered by Dorey et al [96]. For a related recent study of E 6 and E 7 families, see MacKay and Taylor [226].…”

Diagrammatics for F4$F_4$

“…Invariant tensors for classical and exceptional semisimple Lie algebras have been computed diagrammatically by Cvitanović [5], but the approach there is rather different, inspired by the language of Feynman diagrams in quantum field theory. This approach has been further investigated for exceptional Lie algebras in [12, 16, 17].…”