# free Lie algebra

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 $X$.

###### Theorem 1 (Witt).

[1, Thm V.7] $\mathfrak{FL}_{K}\langle X\rangle$ is a free Lie algebra on $X$ 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 $x^{2}$ is not in $\mathfrak{FL}_{K}\langle X\rangle$.

## References

• 1 Nathan Jacobson Lie Algebras, Interscience Publishers, New York, 1962.
Title free Lie algebra FreeLieAlgebra 2013-03-22 16:51:11 2013-03-22 16:51:11 Algeboy (12884) Algeboy (12884) 5 Algeboy (12884) Definition msc 08B20 LieAlgebra UniversalEnvelopingAlgebra PoincareBirkhoffWittTheorem free Lie algebra