Contact Lie algebras, generic stabilisers, and affine seaweeds

Abstract

Let $\mathfrak{q}=LieQ$ be an algebraic Lie algebra of index 1, i.e., a generic $Q$-orbit on ${\mathfrak{q}}^{\ast }$ has codimension 1. We show that the following conditions are equivalent: $\mathfrak{q}$ is contact; a generic $Q$-orbit on ${\mathfrak{q}}^{\ast }$ is not conical; there is a generic stabiliser for the coadjoint action of $\mathfrak{q}$. In addition, if $\mathfrak{q}$ is contact, then the subalgebra $S(\mathfrak{q}{)}_{\mathsf{si}}\subset S(\mathfrak{q})$ generated by symmetric semi-invariants of $\mathfrak{q}$ is a polynomial ring. We study also affine seaweed Lie algebras of type $\mathsf{A}$ and find some contact as well as non-contact examples among them.

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