Rational R-matrices, centralizer algebras and tensor identities for e_6 and e_7 exceptional families of Lie algebras

Details Rational R-matrices, centralizer algebras and tensor identities for e_6 and e_7 exceptional families of Lie algebras Rational R-matrices, centralizer algebras and tensor identities for e_6 and e_7 exceptional families of Lie algebras

2006-08-10

arxiv.org/abs/math/0608248v3

J.Math.Phys.48:103507,2007

Mathematics

Quantum Algebra

24 pages. More minor corrections

Scientific paper

10.1063/1.2779960

We 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.

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