Representations of the exceptional and other Lie algebras with integral eigenvalues of the Casimir operator

TL;DR

This paper extends the understanding of representation decompositions of exceptional Lie algebras, identifying families with integral Casimir eigenvalues up to 9 and providing formulas for their dimensions.

Contribution

It generalizes previous results by describing uniform decompositions for antisymmetrized powers up to n=9 and deriving dimension formulas as functions of the algebra's dimension.

Findings

Families of representations with integer Casimir eigenvalues 5 to 9 identified.

Dimension formulas expressed as functions of the Lie algebra's dimension D.

Potential extension of formulas to broader classes of Lie algebras discussed.

Abstract

The uniformity, for the family of exceptional Lie algebras g, of the decompositions of the powers of their adjoint representations is well-known now for powers up to the fourth. The paper describes an extension of this uniformity for the totally antisymmetrised n-th powers up to n=9, identifying (see Tables 3 and 6) families of representations with integer eigenvalues 5,...,9 for the quadratic Casimir operator, in each case providing a formula (see eq. (11) to (15)) for the dimensions of the representations in the family as a function of D=dim g. This generalises previous results for powers j and Casimir eigenvalues j, j<=4. Many intriguing, perhaps puzzling, features of the dimension formulas are discussed and the possibility that they may be valid for a wider class of not necessarily simple Lie algebras is considered.