# On seaweed subalgebras and meander graphs in type D

## TL;DR

This paper extends the meander graph method for computing the index of seaweed subalgebras to type D Lie algebras, addressing complexities introduced by the branching Dynkin diagram.

## Contribution

It develops a new approach for type D seaweed subalgebras using meander graphs, expanding previous methods from types A, B, and C.

## Key findings

- New phenomena due to the branching Dynkin diagram
- Extension of meander graph approach to type D
- Insights into the structure of seaweed subalgebras in type D

## Abstract

In 2000, Dergachev and Kirillov introduced subalgebras of "seaweed type" in $\mathfrak{gl}_n$ and computed their index using certain graphs, which we call type-${\sf A}$ meander graphs. Then the subalgebras of seaweed type, or just "seaweeds", have been defined by Panyushev (2001) for arbitrary reductive Lie algebras. Recently, a meander graph approach to computing the index in types ${\sf B}$ and ${\sf C}$ has been developed by the authors. In this article, we consider the most difficult and interesting case of type ${\sf D}$. Some new phenomena occurring here are related to the fact that the Dynkin diagram has a branching node.