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Development of a unified tensor calculus for the exceptional Lie algebras

A. J. Macfarlane·Hendryk Pfeiffer
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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. 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.

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A. J. Macfarlane, Hendryk Pfeiffer. 2002-12-16. Development of a unified tensor calculus for the exceptional Lie algebras. https://doi.org/10.1142/s0217751x04017562

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