# Glider representations of chains of semisimple Lie algebras

## TL;DR

This paper introduces and studies glider representations in the context of chains of semisimple Lie algebras, focusing on Verma glider representations and their relation to root systems.

## Contribution

It defines Verma glider representations for chains of semisimple Lie algebras and explores their existence and properties based on root system relations.

## Key findings

- Verma glider representations are related to root system relations.
- Chains of simple Lie algebras of types A, B, C, D are considered.
- Existence of Verma gliders depends on root system compatibility.

## 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 $D$.