PlanetMath: free Lie algebra

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free Lie algebra

(Definition)

Fix a set and a commuative unital ring . A free -Lie algebra $\mathfrak{L}$ on is any Lie algebra together with an injection $\iota:X\rightarrow \mathfrak{L}$ such that for any -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:

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & X\ar[ld]_{\iota}\ar[rd]^{f} & \ \mathfrak{L}\ar[rr]^{\hat{f}} & & \mathfrak{g}. } } \end{xy}$

To construct a free Lie algebra is generally and indirect process. We begin with any free associative algebra $K\langle X\rangle$ on , which can be constructed as the tensor algebra over a free -module with basis . Then $K\langle X\rangle^-$ is a -Lie algebra with the standard commutator bracket 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 .

Theorem 1 (Witt) [1, Thm V.7] $\mathfrak{FL}_K\langle X\rangle$ is a free Lie algebra on and its universal enveloping algebra is $K\langle X\rangle$ .

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 is not in $\mathfrak{FL}_K\langle X\rangle$ .