The nilradical of a seaweed algebra

TL;DR

The paper computes the index of the nilradical of (type-A) seaweed algebras by proving a direct-sum decomposition n(s) ≅ Z(s) ⊕ g^⟂(P_s) and constructing P_s from a modified meander M_p(s). It then derives a combinatorial, edge-weighted meander formula for ind n(s) via a weighted graph M_n(s), linking the index to central components Cen(s) and edge weights, with a simple lower bound in terms of E1(s) and Cen(s). The work also establishes foundational tools to recover posets from meanders and discusses extensions beyond type A and the breadth invariant as future directions. Overall, the results provide concrete, graph-theoretic mechanisms to accessLie algebra invariants of nilradicals in seaweed algebras and suggest avenues for broader generalization and new invariants.

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.

Figures (7)

• Figure 1: The seaweed algebra $\mathfrak{p}\frac{2|4}{1|2|3}$ (left) and its associated meander (right)
• Figure 2: $\mathrm{M}(\mathfrak{s})$ for $\mathfrak{s}=\mathfrak{p}\frac{2|2|3|1|1|3}{4|3|5}$
• Figure 3: $\mathcal{P}=[6]$ with $1,2\prec3\prec4,5,6$ (left) and the nilpotent Lie poset algebra $\mathfrak{g}^{\prec}(\mathcal{P})$ (right)
• Figure 4: $\mathfrak{n}(\mathfrak{s})$ and $\mathcal{P}_\mathfrak{s}$ for $\mathfrak{s}=\mathfrak{p}\frac{2|3|1|2|2}{7|3}$ and $\mathfrak{p}^A\frac{2|3|1|2|2}{7|3}$
• Figure 5: $\mathrm{M}_{\mathfrak{p}}(\mathfrak{s})$ for $\mathfrak{s}=\mathfrak{p}\frac{2|3|1|2|2}{7|3}$ and $\mathfrak{p}^A\frac{2|3|1|2|2}{7|3}$

Theorems & Definitions (39)

• Theorem 1.1
• Lemma 2.1
• proof
• Example 2.2
• Proposition 2.3
• Example 2.4
• Remark 2.5
• Theorem 2.6: Theorem 4, indexnilposet
• Theorem 3.1
• proof