free Lie algebra
Fix a set $X$ and a commuative unital ring $K$. A free $K$-Lie algebra^{} $\U0001d50f$ on $X$ is any Lie algebra together with an injection $\iota :X\to \U0001d50f$ such that for any $K$-Lie algebra $\U0001d524$ and function $f:X\to \U0001d524$ implies the existance of a unique Lie algebra homomorphism^{} $\widehat{f}:\U0001d50f\to \U0001d524$ where $\iota \widehat{f}=f$. This universal mapping property is commonly expressed as a commutative diagram^{}:
$$\text{xymatrix}\mathrm{\&}X\text{ar}{[ld]}_{\iota}\text{ar}{[rd]}^{f}\mathrm{\&}\U0001d50f\text{ar}{[rr]}^{\widehat{f}}\mathrm{\&}\mathrm{\&}\U0001d524.$$ |
To construct a free Lie algebra is generally and indirect process. We begin with any free associative algebra $K\u27e8X\u27e9$ on $X$, which can be constructed as the tensor algebra over a free $K$-module with basis $X$. Then $K{\u27e8X\u27e9}^{-}$ is a $K$-Lie algebra with the standard commutator bracket $[a,b]=ab-ba$ for $a,b\in K\u27e8X\u27e9$.
Now define $\U0001d509{\U0001d50f}_{K}\u27e8X\u27e9$ as the Lie subalgebra of $K{\u27e8X\u27e9}^{-}$ generated by $X$.
Theorem 1 (Witt).
[1, Thm V.7] $\mathrm{F}\mathit{}{\mathrm{L}}_{K}\mathit{}\mathrm{\u27e8}X\mathrm{\u27e9}$ is a free Lie algebra on $X$ and its universal enveloping algebra is $K\mathit{}\mathrm{\u27e8}X\mathrm{\u27e9}$.
It is generally not true that $\U0001d509{\U0001d50f}_{K}\u27e8X\u27e9=K{\u27e8X\u27e9}^{-}$. For example, if $x\in X$ then ${x}^{2}\in K\u27e8X\u27e9$ but ${x}^{2}$ is not in $\U0001d509{\U0001d50f}_{K}\u27e8X\u27e9$.
References
- 1 Nathan Jacobson Lie Algebras, Interscience Publishers, New York, 1962.
Title | free Lie algebra |
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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 |