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

Abstract

We use Cvitanović’s [Group Theory (Princeton University Press, Princeton, NJ, in press) (http://www.nbi.dk/GroupTheory/); Phys. Rev. D 14, 1536 (1976)] diagrammatic techniques to construct the rational solutions of the Yang-Baxter equation [Yang-Baxter Equation in Integrable Systems, edited by M. Jimbo, Advanced Series in Mathematical Physics Vol. 10 (World scientific, Singapore, 1990)] associated with the e6 and e7 families of Lie algebras, and thus explain Westbury’s [J. Phys. A 36, 2857 (2003)] 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 e7 and e6 on their lowest-dimensional representations and (up to threefold) tensor products thereof, giving bases for them and a range of identities satisfied by the algebras’ defining invariant tensors.