Complete filtered Lie algebra

For the convention $F^1\mathfrak g=\mathfrak g$, each quotient $\mathfrak g/F^n\mathfrak g$ is a nilpotent Lie algebra of class at most $n-1$. The complete filtered algebra is therefore an inverse limit of nilpotent Lie algebras and is commonly called pronilpotent.

Some authors define a pronilpotent Lie algebra abstractly as an inverse limit of nilpotent Lie algebras, without choosing one filtration. A chosen complete bracket-compatible filtration is stronger data and is the convention used here.

A series $\sum_{n\geq1}x_n$ with $x_n\in F^n\mathfrak g$ converges: its image in every quotient $\mathfrak g/F^r\mathfrak g$ is a finite sum, and completeness gives a unique compatible limit. In particular, any formal Lie series whose terms of bracket length $n$ lie in $F^n\mathfrak g$ can be evaluated in $\mathfrak g$.

This makes the Baker–Campbell–Hausdorff series convergent when $k$ has characteristic zero, producing the associated BCH group.

A morphism of complete filtered Lie algebras is a continuous Lie-algebra homomorphism $f:\mathfrak g\to\mathfrak h$ satisfying $f(F^n\mathfrak g)\subseteq F^n\mathfrak h$ for every $n$. Such a map commutes with evaluation of convergent formal Lie series.

• A nilpotent Lie algebra $\mathfrak n$, equipped with its lower central series and extended by zeros, is complete filtered.
• If $\mathfrak g$ is any Lie algebra, the completion $\varprojlim_n\mathfrak g/\gamma_n(\mathfrak g)$ of its lower central series is pronilpotent when the transition maps and brackets satisfy the indicated separatedness conditions.

1. Nicolas Bourbaki, Lie Groups and Lie Algebras, Chapters 1–3, Springer, 1989. Publisher record. Relevant: Chapter II, §§6–7 on formal Lie series and complete filtered algebras.
2. Jean-Pierre Serre, Lie Algebras and Lie Groups, Lecture Notes in Mathematics 1500, Springer, 1992. Publisher record. Relevant: Part II on the Campbell–Hausdorff formula and formal groups.