-2
$\begingroup$

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?

Improve this question
$\endgroup$
4
  • 2
    $\begingroup$ Isn't it quite a trivial Lie algebra, given that the central series stabilizes at its second point (for $n\geq 5$) ? $\endgroup$
    – darij grinberg
    Commented Apr 9, 2011 at 5:45
  • $\begingroup$ 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. $\endgroup$
    – Dr Shello
    Commented Apr 9, 2011 at 6:03
  • 3
    $\begingroup$ 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. $\endgroup$
    – Ben Webster ♦
    Commented Apr 9, 2011 at 6:47
  • 3
    $\begingroup$ 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) $\endgroup$
    – Tobias Kildetoft
    Commented Apr 9, 2011 at 9:19

2 Answers 2

Reset to default
4
$\begingroup$

For any $n>1$, the lower central series for the symmetric group is $S_n > A_n\geq A_n \geq A_n \geq \cdots$, 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 $\mathbb{F}_p$.

Improve this answer
$\endgroup$
3
  • $\begingroup$ 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... $\endgroup$
    – Dr Shello
    Commented Apr 9, 2011 at 15:51
  • $\begingroup$ 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". $\endgroup$
    – S. Carnahan ♦
    Commented Apr 9, 2011 at 16:06
  • 1
    $\begingroup$ @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. $\endgroup$
    – M T
    Commented Apr 9, 2011 at 19:37
1
$\begingroup$

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.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .