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

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

2007-07-24

arxiv.org/abs/0707.3597v1

Mathematics

Representation Theory

Scientific paper

In this paper, we study the set of $\Delta$-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 $\Delta$-dimension vector, there is a unique (up to isomorphism) exceptional $\Delta$-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.

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