Rationally varying polarizing subalgebras in nilpotent Lie algebras

Abstract:

Let $\mathfrak {N}$ be a nilpotent Lie algebra, with (vector space) dual ${\mathfrak {N}^{*}}$. We construct a map $l \mapsto {\mathfrak {M}_l}$ from a Zariski-open subset of ${\mathfrak {N}^{*}}$ to the set of subalgebras of $\mathfrak {N}$ such that ${\mathfrak {M}_l}$ varies rationally with $l$, ${\mathfrak {M}_l}$ is polarizing for $l$, $({\text {Ad}}x){\mathfrak {M}_l} = {\mathfrak {M}_{l’}}(l’ = {\text {A}}{{\text {d}}^ {*} }x \cdot l)$ for all $x \in N$, and $un{k_\infty }(l) \subseteq {\mathfrak {M}_l} \subseteq un{k_\infty }(l)$, where $un{k_\infty }(l)$ and $un{k_\infty }(l)$ are the canonical subalgebras introduced by R. Penney.