# Hall Basis of a Free Lie Algebra [closed]

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 KohlDec 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

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.
I think your first example should be $[x_1,x_1]$. –  Tom Church May 25 '11 at 15:34
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