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
Canonical name	FreeLieAlgebra
Date of creation	2013-03-22 16:51:11
Last modified on	2013-03-22 16:51:11
Owner	Algeboy (12884)
Last modified by	Algeboy (12884)
Numerical id	5
Author	Algeboy (12884)
Entry type	Definition
Classification	msc 08B20
Related topic	LieAlgebra
Related topic	UniversalEnvelopingAlgebra
Related topic	PoincareBirkhoffWittTheorem
Defines	free Lie algebra