Development of a unified tensor calculus for the exceptional Lie algebras

TL;DR

This paper develops a unified tensor calculus for exceptional Lie algebras, systematically analyzing tensors at the n=2 and n=3 levels to facilitate their representation theory and identities.

Contribution

It introduces a systematic tensor calculus for exceptional Lie algebras, including explicit analysis at n=2 and initial results at n=3, advancing understanding of their tensor structures.

Findings

Explicit description of tensors at n=2 stage

Dimension and Casimir eigenvalues for ad^3 constituents

Special analysis of d_4 within the family F

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.