Title: Glider representations of chains of semisimple Lie algebras,
Abstract: We start the study of glider representations in the setting of semisimple Lie
algebras. A glider representation is defined for some positively filtered ring
$FR$ and here we consider the right bounded algebra filtration
$FU(\mathfrak{g})$ on the universal enveloping algebra $U(\mathfrak{g})$ of
some semisimple Lie algebra $\mathfrak{g}$ given by a fixed chain of semisimple
sub Lie algebras $\mathfrak{g}_1 \subset \mathfrak{g}_2 \subset \ldots \subset
\mathfrak{g}_n = \mathfrak{g}$. Inspired by the classical representation
theory, we introduce so-called Verma glider representations. Their existence is
related to the relations between the root systems of the appearing Lie algebras
$\mathfrak{g}_i$. In particular, we consider chains of simple Lie algebras of
the same type $A,B,C$ and $D$.