# Motivation for filtrations on a group and associated Lie algebras

In his Lie algebras and Lie groups, the first non-trivial example of a Lie algebra that Serre gives is a graded Lie algebra associated with a filtration on a group. To me this construction looks both extremely daring and completely artificial, I'm yet to see a single example of it outside the Serre's book. I also assume that they are very different from the ones over archimedian fields that I know and love from differential geometry because of their $\mathbb{R}$-grading.

How do such Lie algebras arise in the wild? Are they useful for something or just interesting for their own sake?

• @Dylan, I don't know. What? – Alexei Averchenko Dec 5 '11 at 14:59
• Apparently you can use this construction to define free Lie algebras: en.wikipedia.org/wiki/Free_Lie_algebra#Hall_sets – Qiaochu Yuan Dec 5 '11 at 15:03
• @Qiaochu, oh, it's even in the book; silly me :) – Alexei Averchenko Dec 5 '11 at 15:12
• please also search for "Lie methods in growth" – niyazi Dec 5 '11 at 15:17

Let $G$ be a finite $p$-group. Let $(\Gamma_i)$ be a certain descending series with elementary abelian quotients. Let $A=\operatorname{gr}(\mathbb{F}_pG)$ be the graded algebra associated to the radical filtration of $\mathbb{F}_p(G)$, that is, $A$ is the graded algebra with $i$th piece $A_i = J^i/J^{i+1}$ where $J=\operatorname{rad}\mathbb{F}_p(G)$, and whose multiplication is inherited from $\mathbb{F}_p(G)$. Then $A$ is isomorphic to the restricted enveloping algebra of the $p$-restricted Lie algebra associated to the series $\Gamma_i$.
This fact is due to Quillen; it is a high-tech way of expressing Jennings' theorem. Details can be found in Benson's Representations and Cohomology volume 1. It is useful because it gives information on the radical series that would otherwise be hard to obtain, and also gives a good deal of cohomological information about the graded algebra $A$ (e.g. its ordinary cohomology ring is finitely generated over the subring generated by elements of degree at most two). In certain cases (when $\mathbb{F}_p(G)$ is "tightly graded" by its radical filtration) $A \cong \mathbb{F}_pG$ and you obtain cohomological information about $G$ itself.