Lie algebras of triangular polynomial derivations and an isomorphism criterion for their Lie factor algebras

We make a detailed study of the Lie algebras $\mathfrak{u}_n$, $n\geqslant 2$, of triangular polynomial derivations, their injective limit $\mathfrak{u}_\infty$, and their completion $\widehat{\mathfrak{u}}_\infty$. We classify the ideals of $\mathfrak{u}_n$ (all of which are characteristic ideals) and use this classification to give an explicit criterion for Lie factor algebras of $\mathfrak{u}_n$ and $\mathfrak{u}_m$ to be isomorphic. We introduce two new dimensions for (Lie) algebras and their modules: the central dimension $\operatorname{c.dim}$ and the uniserial dimension $\operatorname{u.dim}$, and show that $\operatorname{c.dim}(\mathfrak{u}_n)=\operatorname{u.dim}(\mathfrak{u}_n) =\omega^{n-1}+\omega^{n-2}+\dots+\omega +1$ for all $n\geqslant 2$, where $\omega$ is the first infinite ordinal. Similar results are proved for the Lie algebras $\mathfrak{u}_\infty$ and $\widehat{\mathfrak{u}}_\infty$. In particular, $\operatorname{u.dim}(\mathfrak{u}_\infty)=\omega^\omega$ and $\operatorname{c.dim}(\mathfrak{u}_\infty)=0$.