## Lie algabra of symmetric group

It's easy to see that the descending central series of a group induces a graded Lie algebra .(see for example Serre's Harvard lectures or Magnus-Solitar book). I think in general this can be complicated, but this should be well-known:

What is the structure of this Lie algebra for the symmetric group?

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Isn't it quite a trivial Lie algebra, given that the central series stabilizes at its second point (for $n\geq 5$) ? – darij grinberg Apr 9 2011 at 5:45
The only example I know (from book above) is for the free groups where it is a theorem that the Lie algebra is free (not too hard proof, but not trivial). I'm trying to find other examples with known ansers; I'm not sure where this kind of thing is found. – Dr Shello Apr 9 2011 at 6:03
This construction only works well for nilpotent groups. If you think a bit about what the descending central series of the symmetric group is, you will see that this is not a very good example. – Ben Webster Apr 9 2011 at 6:47
Also, unless you have some extra criteria on the group, it will generally only be a Lie ring (there is no reason to expect the quotients in the series to be vector spaces over some field) – Tobias Kildetoft Apr 9 2011 at 9:19

## 2 Answers

For any $n>1$, the lower central series for the symmetric group is $S_n > A_n > A_n > A_n > \cdots$, so the Lie ring formed by the sum of successive quotients is the group $\mathbb{Z}/2\mathbb{Z}$, equipped with the Lie bracket that is identically zero.

If you want to gain intuition for this construction with finite groups, I suggest you consider nilpotent groups, since their lower central series actually reach the trivial group. For example, many $p$-groups will yield nonabelian Lie algebras over $\mathbb{F}_p$.

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I see, thanks. Looks like I asked this question (much!) too hastily. Does anyone have any reference recommendations for these DCS Lie algebras? None of the books I've been poking around in have much... – Dr Shello Apr 9 2011 at 15:51
This is not my field, but there seem to be many relevant results that appear when you search for terms like "Lie ring" and "nilpotent group". – S. Carnahan Apr 9 2011 at 16:06
@Dr Shello: you will get interesting examples with $p$-groups $G$ of exponent $p$. Then Quillen's version of Jennings' theorem tells you the restricted enveloping algebra of the Lie algebra that arises is isomorphic to the graded algebra associated to the radical filtration of $\mathbb{F}_pG$. Quillen's paper is J.Alg 10 (1968) pp.411-418, his result is also in Benson's book Representations and Cohomology I. – mt Apr 9 2011 at 19:37

For a well-known infinite example, there is a result, due to Labute ("On the descending central series of groups with a single defining relation", J. Algebra 14 (1970), 16--23) which asserts that the Lie ring associated to a one-relator group can be presented as a Lie ring with a single defining relator.

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