Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
    
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 '11 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 '11 at 6:03
1  
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 '11 at 6:47
2  
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 '11 at 9:19
add comment

2 Answers

up vote 3 down vote accepted

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 $\mathbb{F}_p$.

share|improve this answer
    
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 '11 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 '11 at 16:06
1  
@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. –  m_t Apr 9 '11 at 19:37
add comment

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.