What is the definition of polynilpotent Lie algebras?

Question

I am looking for definition of polynilpotent Lie algebras. Is there any equivalent concept for that?

Answer 1

Usually poly-P means that there is a subnormal series (or normal series; here it doesn't matter) in which all subquotients have Property P.

When P is "abelian" or "solvable", clearly poly-P is the same as solvable. Since nilpotent is between solvable and abelian, poly-nilpotent would just be the same as solvable.

So the definition has no interest. Unless one counts the number of steps. Then being $n$-step poly-nilpotent, i.e. polynilpotent with a subnormal series of length $n$ has a reasonable meaning. For instance for $n=2$ it is called meta-nilpotent.

Over a field, every solvable finite-dimensional Lie algebra is nilpotent-by-abelian, hence 2-step polynilpotent.

In infinite dimension one can construct solvable Lie algebra of arbitrary large minimal step of polynilpotency.

Answer 2

If $L$ is a Lie algebra then by iteration the lower central series is defined $L^1=L$; $L^{i+1}=[L; L^{i}]$; $i = 1,2, \dots$. Now $L$ is called nilpotent of class $s$ iff $L^{s+1}=0$ and $L^{s} \neq 0$. All Lie algebras of class $s$ form a variety $N_s$. So, $L$ is called polynilpotent with tuple $(s_q, \dots, s_2, s_1)$ iff there is a chain of ideals $0=L_{q+1} \subset L_{q} \subset \dots L_{2} \subset L_{1}=L $ with $L_{i} / L_{i+1} \in N_{s_{i}}$.