Exceptional representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras

Jensen, Bernt Tore; Su, Xiuping; Yu, Rupert W. T.

Exceptional representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras

Authors

Jensen, Bernt Tore

Su, Xiuping

Yu, Rupert W. T.

Description

In this paper, we study the set of $Δ$-filtered modules of quasi-hereditary algebras arising from quotients of the double of quivers of type $A$. Our main result is that for any fixed $Δ$-dimension vector, there is a unique (up to isomorphism) exceptional $Δ$-filtered module. We then apply this result to show that there is always an open adjoint orbit in the nilpotent radical of a seaweed Lie algebra in $\mathrm{gl}_{n}(\field)$, thus answering positively in this $\mathrm{gl}_{n}(\field)$ case to a question raised independently by Michel Duflo and Dmitri Panyushev. An example of a seaweed Lie algebra in a simple Lie algebra of type $E_{8}$ not admitting an open orbit in its nilpotent radical is given.