Filtrations in semisimple Lie algebras, I

In this paper, we study the maximal bounded $\mathbb{Z}$-filtrations of a complex semisimple Lie algebra $L$. Specifically, we show that if $L$ is simple of classical type $A_n$, $B_n$, $C_n$ or $D_n$, then these filtrations correspond uniquely to a precise set of linear functionals on its root space. We obtain partial, but not definitive, results in this direction for the remaining exceptional algebras. Maximal bounded filtrations were first introduced in the context of classifying the maximal graded subalgebras of affine Kac-Moody algebras, and the maximal graded subalgebras of loop toroidal Lie algebras. Indeed, our main results complete this classification in most cases. Finally, we briefly discuss the analogous question for bounded filtrations with respect to other Archimedean ordered groups.