Describing certain Lie algebra orbits via polynomial equations

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).