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. 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.

27 pages, LaTeX 2e

References in corpus (2)

• Lie algebra and invariant tensor technology for g2
• Representations of the exceptional and other Lie algebras with integral eigenvalues of the Casimir operator

Cited by in corpus (1)

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