Existence of Richardson elements in seaweed Lie algebras of type $\mathbb{B}$, $\mathbb{C}$ and $\mathbb{D}$

TL;DR

This paper proves the existence of Richardson elements in seaweed Lie algebras of types B, C, and D, completing the classification for finite types and using quiver theory for verification.

Contribution

It extends the existence results of Richardson elements to all seaweed Lie algebras of types B, C, and D, using decomposition and quiver representation techniques.

Findings

Richardson elements exist for all seaweed Lie algebras of types B, C, and D.

A sufficient condition for existence is established and verified.

Two special cases are identified where direct proofs are provided.

Abstract

Seaweed Lie algebras are a natural generalisation of parabolic subalgebras of reductive Lie algebras. The well-known Richardson Theorem says that the adjoint action of a parabolic group has a dense open orbit in the nilpotent radical of its Lie algebra \cite{richardson}. We call elements in the open orbit Richardson elements. In \cite{JSY} together with Yu, we generalized Richardson's Theorem and showed that Richardson elements exist for seaweed Lie algebras of type $\mathbb{A}$. Using GAP, we checked that Richardson elements exist for all exceptional simple Lie algebras except $\mathbb{E}_8$, where we found a counterexample. In this paper, we complete the story on Richardson elements for seaweeds of finite type, by showing that they exist for any seaweed Lie algebra of type $\mathbb{B}$, $\mathbb{C}$ and $\mathbb{D}$. By decomposing a seaweed into a sum of subalgebras and analysing their stabilisers, we obtain a sufficient condition for the existence of Richarson elements. The sufficient condition is then verified using quiver representation theory. More precisely, using the categorical construction of Richardson elements in type $\mathbb{A}$, we prove that the sufficient condition is satisfied for all seaweeds of type $\mathbb{B}$, $\mathbb{C}$ and $\mathbb{D}$, except in two special cases, where we give a directproof.