Development of a Unified Tensor Calculus for the Exceptional Lie Algebras

Abstract

The uniformity of the decomposition law, for a family F of Lie algebras which includes the exceptional Lie algebras, of the tensor powers ad ⊗n of their adjoint representations ad is now well-known. This paper uses it to embark on the development of a unified tensor calculus for the exceptional Lie algebras. It deals explicitly with all the tensors that arise at the n = 2 stage, obtaining a large body of systematic information about their properties and identities satisfied by them. Some results at the n = 3 level are obtained, including a simple derivation of the the dimension and Casimir eigenvalue data for all the constituents of ad ⊗3 . This is vital input data for treating the set of all tensors that enter the picture at the n = 3 level, following a path already known to be viable for a 1 ∈ F .The special way in which the Lie algebra d 4 conforms to its place in the family F alongside the exceptional Lie algebras is described.

Deformation Quantization on the Closure of Minimal Coadjoint Orbits

“…Here we use some results of [63]. Let κ denote the Killing form, and κ i j the matrix elements of the inverse matrix.…”

“…The following table (see [63]) gives the eigenvalues l i = l 1 , l 2 , l 3 associated with the representations I, R 2 , R 3 . In all cases, we have l 1 = 1, and l 2 + l 3 = − We turn to Equation (24).…”

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

“…(Note that such uniformity suggests an extension of Deligne's conjecture [21], about the uniformity of decomposition of g ⊗r , to Yangians.) Although both conventional [22] and diagrammatic [3] techniques for the adjoint representation of the e 8 series (the latter as advocated in [23,24]) are well-developed, we have not yet been able to extend them to this reducible representation. Such a treatment of the R-matrix remains, however, highly desirable, as a step towards explaining the remarkable appearance of spectra associated with the algebras of the e 8 series in the q-state Potts model S-matrix [25,26].…”