Describing certain Lie algebra orbits via polynomial equations

TL;DR

This paper characterizes the orbit closures of specific 3-dimensional Lie algebras, including the Heisenberg algebra, using polynomial equations, providing a new algebraic approach to understanding their degenerations.

Contribution

It offers explicit polynomial descriptions of orbit closures for certain Lie algebras over any infinite field, enabling a novel algebraic method to study their degenerations.

Findings

Explicit polynomial equations for orbit closures of $rak{h}_3$ and $rak{g}$.

A new algebraic approach to classify degenerations of these Lie algebras.

Application over arbitrary infinite fields.

Abstract

Let $\mathfrak{h}_3$ be the Heisenberg algebra and let $\mathfrak g$ be the 3-dimensional Lie algebra having $[e_1,e_2]=e_1\,(=-[e_2,e_1])$ as its only non-zero commutation relations. We describe the closure of the orbit of a vector of structure constants corresponding to $\mathfrak{h}_3$ and $\mathfrak g$ respectively as an algebraic set giving in each case a set of polynomials for which the orbit closure is the set of common zeros. Working over an arbitrary infinite field, this description enables us to give an alternative way, using the definition of an irreducible algebraic set, of obtaining all degenerations of $\mathfrak{h}_3$ and $\mathfrak g$ (the degeneration from $\mathfrak g$ to $\mathfrak{h}_3$ being one of them).