free Lie algebraFreeLieAlgebra2007-03-21 16:41:552007-03-22 10:57:05Definitionfree algebrafree Lie algebra\usepackage{latexsym}
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Fix a set $X$ and a commuative unital ring $K$. A free $K$-Lie algebra $\mathfrak{L}$ on $X$ is any Lie algebra together with an injection $\iota:X\rightarrow \mathfrak{L}$
such that for any $K$-Lie algebra $\mathfrak{g}$ and function
$f:X\rightarrow \mathfrak{g}$ implies
the existance of a unique Lie algebra homomorphism $\hat{f}:\mathfrak{L}\rightarrow \mathfrak{g}$
where $\iota\hat{f}=f$. This universal mapping property is commonly expressed
as a commutative diagram:
\[\xymatrix{
& X\ar[ld]_{\iota}\ar[rd]^{f} & \\
\mathfrak{L}\ar[rr]^{\hat{f}} & & \mathfrak{g}.
}\]
To construct a free Lie algebra is generally and indirect process.
We begin with any free associative algebra $K\langle X\rangle$ on $X$,
which can be constructed as the tensor algebra over a free $K$-module
with basis $X$. Then $K\langle X\rangle^-$ is a $K$-Lie algebra with the
standard commutator bracket $[a,b]=ab-ba$ for $a,b\in K\langle X\rangle$.
Now define $\mathfrak{FL}_K\langle X\rangle$ as the Lie subalgebra of $K\langle X\rangle^-$
generated by $X$.
\begin{thm}[Witt]\cite[Thm V.7]{Jacobson}
$\mathfrak{FL}_K\langle X\rangle$ is a free Lie algebra on $X$ and its universal
enveloping algebra is $K\langle X\rangle$.
\end{thm}
It is generally not true that $\mathfrak{FL}_K\langle X\rangle=K\langle X\rangle^-$. For example, if $x\in X$ then $x^2\in K\langle X\rangle$ but
$x^2$ is not in $\mathfrak{FL}_K\langle X\rangle$.
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\bibitem{Jacobson}
Nathan Jacobson \emph{Lie Algebras}, Interscience Publishers, New York, 1962.
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