A new characterization of the exceptional Lie algebras

Abstract

For a simple Lie algebra, over $\mathbb{C}$, we consider the weight which is the sum of all simple roots and denote it $\tilde{\alpha}$. We formally use Kostant's weight multiplicity formula to compute the "dimension" of the zero-weight space. In type $A_r$, $\tilde{\alpha}$ is the highest root, and therefore this dimension is the rank of the Lie algebra. In type $B_r$, this is the defining representation, with dimension equal to 1. In the remaining cases, the weight $\tilde{\alpha}$ is not dominant and is not the highest weight of an irreducible finite-dimensional representation. Kostant's weight multiplicity formula, in these cases, is assigning a value to a virtual representation. The point, however, is that this number is nonzero if and only if the Lie algebra is classical. This gives rise to a new characterization of the exceptional Lie algebras as the only Lie algebras for which this value is zero.