Contact Lie algebras, generic stabilisers, and affine seaweeds

Abstract

Let $\mathfrak q=Lie Q$ be an algebraic Lie algebra of index 1, i.e., a generic $Q$-orbit on $\mathfrak q^*$ has codimension 1. We show that the following conditions are equivalent: $\mathfrak q$ is contact; a generic $Q$-orbit on $\mathfrak q^*$ 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 $S(\mathfrak q)_{\sf si}\subset S(\mathfrak q)$ generated by symmetric semi-invariants of $\mathfrak q$ is a polynomial ring. We study also affine seaweed Lie algebras of type ${\sf A}$ and find some contact as well as non-contact examples among them.