Contact Lie algebras, generic stabilisers, and affine seaweeds

TL;DR

This paper explores the properties of contact Lie algebras of index 1, establishing their equivalence with certain geometric conditions and analyzing affine seaweed Lie algebras for contact structures and invariants.

Contribution

It proves the equivalence of contact structure, orbit conicality, and stabilizer existence for index 1 Lie algebras, and studies contact properties in affine seaweed Lie algebras.

Findings

Contact Lie algebras of index 1 have polynomial rings of semi-invariants.

Equivalence between contact structure, orbit conicality, and stabilizer existence.

Identification of contact and non-contact affine seaweed Lie algebras.

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.