On seaweed subalgebras and meander graphs in type C

Dmitri I. Panyushev and Oksana S. Yakimova

Vol. 285 (2016), No. 2, 485–499

DOI: 10.2140/pjm.2016.285.485

Abstract

In 2000, Dergachev and Kirillov introduced subalgebras of “seaweed type” in ${\mathfrak{g}\mathfrak{l}}_n$ (or ${\mathfrak{s}\mathfrak{l}}_n$) and computed their index using certain graphs. In this article, those graphs are called type-A meander graphs. Then the subalgebras of seaweed type, or just “seaweeds”, were defined by Panyushev (2001) for arbitrary simple Lie algebras. Namely, if ${\mathfrak{p}}_1,{\mathfrak{p}}_2\subset \mathfrak{g}$ are parabolic subalgebras such that ${\mathfrak{p}}_1+{\mathfrak{p}}_2=\mathfrak{g}$, then $\mathfrak{q}={\mathfrak{p}}_1\cap {\mathfrak{p}}_2$ is a seaweed in $\mathfrak{g}$. If ${\mathfrak{p}}_1$ and ${\mathfrak{p}}_2$ are “adapted” to a fixed triangular decomposition of $\mathfrak{g}$, then $\mathfrak{q}$ is said to be standard. The number of standard seaweeds is finite. A general algebraic formula for the index of seaweeds was proposed by Tauvel and Yu (2004) and then proved by Joseph (2006).

In this paper, elaborating on the “graphical” approach of Dergachev and Kirillov, we introduce the type-C meander graphs, i.e., the graphs associated with the standard seaweed subalgebras of ${\mathfrak{s}\mathfrak{p}}_{2n}$, and give a formula for the index in terms of these graphs. We also note that the very same graphs can be used in the case of the odd orthogonal Lie algebras.

Recall that $\mathfrak{q}$ is called Frobenius if the index of $\mathfrak{q}$ equals $0$. We provide several applications of our formula to Frobenius seaweeds in ${\mathfrak{s}\mathfrak{p}}_{2n}$. In particular, using a natural partition of the set ${\mathcal{ℱ}}_n$ of standard Frobenius seaweeds, we prove that $\#{\mathcal{ℱ}}_n$ strictly increases for the passage from $n$ to $n+1$. The similar monotonicity question is open for the standard Frobenius seaweeds in ${\mathfrak{s}\mathfrak{l}}_n$, even for the passage from $n$ to $n+2$.

Keywords

index of Lie algebra, Frobenius Lie algebra

Mathematical Subject Classification 2010

Primary: 17B08

Secondary: 17B20

Milestones

Received: 29 December 2015

Revised: 5 May 2016

Accepted: 28 May 2016

Published: 21 November 2016