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Since a basis for a Free Lie Algebra $\cal{FL}$ with n generators $x_1,\ldots x_r$ can be described by the elements $\{ [\ldots[x_{i_1},x_{i_2}]x_{i_2}\ldots]x_{i_m}] , 1 \leq x_{i_j} \leq r , 1 \leq j \leq m , m \in \mathbb{N} \}$ , what is the point of using the Hall basis for $\cal{FL}$? Both basis generate the same graded algebra and the first basis is simpler than the Hall basis .

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closed as off-topic by Ricardo Andrade, Daniel Moskovich, Jack Huizenga, Andrey Rekalo, Stefan Kohl Dec 1 '13 at 9:22

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With no restrictions on the indices, that's not a basis. –  Qiaochu Yuan May 25 '11 at 15:13

1 Answer 1

Your elements don't form a basis. You haven't given any conditions on the indices $i_j$, so $[x_1,x_1]$ is in your list, but it is zero. Now consider words of length three. Among these are $[[x_1,x_2],x_3]$, $[[x_2,x_3],x_1]$ and $[[x_3,x_1],x_2]$. The sum of these three is zero by the Jacobi identity, so they are not linearly independent. You might want to insist that the list of indices $i_1,\dotsc,i_r$ is strictly increasing, but then $[[x_2,x_3],x_1]$ would not be in the span.

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I think your first example should be $[x_1,x_1]$. –  Tom Church May 25 '11 at 15:34
    
@Tom: fixed, sorry –  Neil Strickland May 25 '11 at 15:43
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...and if you start considering more and more complicated instances of these issues, you start to see why Hall bases are actually surprisingly simple things in view of the problem they solve! –  Mariano Suárez-Alvarez May 25 '11 at 17:54
    
Call $LNB(w)$ for left normed bracketing dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/viewArticle/1014 of the word $w=x_1x_2\cdots x_n$ the element $[..[[x_1,x_2],\cdots x_n]$ (it is indeed $n$ times the Dynkin projection of the word $w$). Up to my knowledge, it is still open to have a elegant or efficient description of a set of words $S$ for which the $\{[w]\}_{w\in S}$ form a basis of the free lie algebra. –  Duchamp Gérard H. E. Aug 7 '13 at 8:38
    
Call $LNB(w)$ for left normed bracketing dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/viewArticle/1014 of the word $w=x_1x_2\cdots x_n$ the element $[..[[x_1,x_2],\cdots x_n]$ (it is indeed $n$ times the Dynkin projection of the word $w$). Up to my knowledge, it is still open to have an elegant or efficient description of a set of words $S$ for which the $\{LNB(w)\}_{w\in S}$ form a basis of the free lie algebra. –  Duchamp Gérard H. E. Aug 7 '13 at 8:46

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