Bracket ideals and Hilbert polynomial of filiform Lie algebras

For a complex finite-dimensional filiform Lie algebra $\mathfrak g$, we first study the bifiltration given by the bracket ideals $[C^k\mathfrak g,C^\ell\mathfrak g]$ and then the behavior of its associated bivariate Hilbert polynomial. This behavior depends in particular on two numerical invariants that measure, on one hand, certain properties of the centralizers in $\mathfrak g$ of the ideals in the lower central sequence and, on the other hand, the dimension of the largest abelian ideal that appears in the lower central series. We give examples proving that the Hilbert polynomial can distinguish isomorphism classes of filiform Lie algebras that cannot be distinguished by the two aforementioned invariants.