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

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