Contact Lie algebras, generic stabilisers, and affine seaweeds
Oksana Yakimova
TL;DR
The paper characterises when index-1 Lie algebras are contact, showing this is equivalent to the existence of a generic stabiliser for the coadjoint action and to the associated generic orbit being non-conical. It develops a framework linking contact structures to invariants and semi-invariants, proving that for contact algebras the ring of semi-invariants is a polynomial ring and that canonical truncation preserves these polynomial structures, with implications for current algebras. It applies these ideas to seaweed subalgebras (bi-parabolics) and, in particular, to affine seaweed subalgebras of type A, deriving index formulas and identifying both contact and non-contact examples; notably, index-1 affine seaweeds arising from even parity data are often quasi-reductive and contact, while many odd-parameter cases fail. The work connects broader concepts of quasi-reductivity, stability, and SK theory to provide a general route for recognizing contact structures in non-reductive contexts, including explicit criteria for semi-direct products and concrete index computations for affine seaweeds. Overall, the results extend known classifications of contact seaweeds beyond reductive settings and highlight rich interactions between invariants, semi-invariants, and the geometry of coadjoint orbits in index-1 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.