The nilradical of a seaweed algebra

TL;DR

This paper provides an explicit formula for the index of nilradicals of seaweed subalgebras of general linear and special linear Lie algebras, linking combinatorial graph structures to algebraic properties.

Contribution

It introduces a new edge-weighted meander graph approach to compute the index of these nilradicals and describes their decomposition into simpler algebraic components.

Findings

Explicit index formula for nilradicals using weighted meanders

Decomposition of nilradicals into center and nilpotent poset algebra

Procedure to recover underlying poset from meander graph

Abstract

Seaweed subalgebras of $\mathfrak{gl}(n,\mathbb{C})$ and $\mathfrak{sl}(n,\mathbb{C})$ are combinatorially defined matrix Lie algebras whose index admits a closed-form description in terms of an associated graph called a meander. In this paper, we study the nilradicals of these algebras with our main result establishing an explicit formula for their index in terms of an edge-weighted variation of the meander. We further prove that each such nilradical decomposes as a direct sum of the center of the seaweed subalgebra with a nilpotent Lie poset algebra, and we provide a meander-theoretic procedure for recovering the underlying poset.