Seaweed algebras

TL;DR

This paper introduces seaweed algebras, explaining how their index can be computed using combinatorial methods involving meander graphs, making the process accessible to a broader audience.

Contribution

It provides a survey of seaweed algebras, including new results and a simplified combinatorial approach to computing their index, with minimal reliance on advanced Lie theory.

Findings

Index computation reduces to a combinatorial formula

Introduction of meander graphs for algebra analysis

New results on seaweed algebra properties

Abstract

The index of a Lie algebra is an important algebraic invariant, but it is notoriously difficult to compute. However, for the suggestively-named seaweed algebras, the computation of the index can be reduced to a combinatorial formula based on the connected components of a "meander": a planar graph associated with the algebra. Our index analysis on seaweed algebras requires only basic linear and abstract algebra. Indeed, the main goal of this survey-type article is to introduce a broader audience to seaweed algebras with minimal appeal to specialized language and notation from Lie theory. This said, we present several results that do not appear elsewhere and do appeal to more advanced language in the Introduction to provide added context.