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I'm working on question 4.8 on page 36 of Erdmann and Wildon's book called Introduction to Lie Algebras. The question is as follows:

Let $L$ be a Lie algebra over a field $F$, such that $[a,b],b]=0$ for all $a,b \in L$.

(i) If $Char(F) \neq 3$, then show that $L^3=0$;

(ii) if $F$ has characteristic 3 then show that $L^4=0$. Here $L^k=[L L^{k-1}]$ and $L^1=[L L]$.

Now to the key is to look at $[[a,b+c],b+c]$ and deduce that $[[a,c],b]=-[[a,b],c]$, and then use Jacobi to show that $[[a,b],c]=0$ for all $a,b,c \in L$. But doesn't this mean that $L^2=0$?. I'm just a little worried that I seem to have proven something stronger than the question asks, which makes me uneasy. Similarly for part (ii) I think the proof shows $L^3=0$.

So I was wondering if there is something I am overlooking.

Thank you

Chris Birkbeck
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    Why do you think it means $L^2 = 0$? You have shown that $[[a, b], c] = 0$ for all $a, b, c \in L$, not that $[a, b] = 0$ for all $a, b \in L$. – Jim Apr 05 '13 at 17:34
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    My first guess was that the indexing is off by one, because I find it a bit strange that $L^1\neq L$. But according to that definition this may well be the case for many $L$ :-) @Jim: See the strange definition of $L^k$ given in item ii) ! – Jyrki Lahtonen Apr 05 '13 at 17:34
  • Yeah well thats the definition he gives in the book for $L^k$ – Chris Birkbeck Apr 05 '13 at 18:00
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    When talking about lower central series, I believe it to be pretty standard to define it as $L_1 = L$ and $L_k=[L,L_{k-1}]$ for $k\geq2$. Do not get confused with the derived series where $L^1= [L,L]$ and $L^k=[L^{k-1},L^{k-1}]$ for $k\geq2$. –  Apr 09 '13 at 19:20
  • i.e note their differing starting points! $L^1 = L_2$ –  Apr 09 '13 at 19:22
  • @ChrisBirkbeck The book is by Erdmann and Wildon, and Erdmann is a she. – Matthew Towers Nov 20 '14 at 10:44
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    @mt_ Thanks, I've edited the question now, but I can't see how to edit my comment. At least I know now for future reference. – Chris Birkbeck Nov 21 '14 at 11:46

1 Answers1

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This CW answer intends to remove this question from the Unanswered queue.

As already noted in the comments, you are correct. It seems like the authors themselves got confused with the definitions.

Julian Kuelshammer
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