Ad-Nilpotent Subalgebras of \(\mathbb{Z}_n\)-graded Lie algebras

Let \(\mathfrak{g}\) be a (semi-) simple Lie algebra over \(\mathbb{C}\) with a \(\mathbb{Z}_n\)-grading. In this talk, I will discuss certain subspaces of the positively graded part of \(\mathfrak{g}\), which have a nice classification using certain subsets of the set of weights of \(\mathfrak{g}_+\) as a \(\mathfrak{g}_0\)-module. This talk is based on work currently in preparation with A. Sandler.