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

Abstract

The uniformity, for the family of exceptional Lie algebras g ,o f the decompositions of the powers of their adjoint representations is now well known for powers up to four. The paper describes an extension of this uniformity for the totally antisymmetrized nth powers up to n = 9, identifying (see tables 3 and 6 )f amilies of representations with integer eigenvalues 5, ..., 9 for the quadratic Casimir operator, in each case providing a formula (see equations (11)–(15)) for the dimensions of the representations in the family as af unction of D = dim g .T his generalizes previous results for powers j and Casimir eigenvalues j, j 4. Many intriguing, perhaps puzzling, features of the dimension formulae are discussed and the possibility that they may be valid for a wider class of not necessarily simple Lie algebras is considered.