free Lie algebra

Fix a set X and a commuative unital ring K. A free K-Lie algebraMathworldPlanetmath 𝔏 on X is any Lie algebra together with an injection ι:X𝔏 such that for any K-Lie algebra 𝔤 and function f:X𝔤 implies the existance of a unique Lie algebra homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath f^:𝔏𝔤 where ιf^=f. This universal mapping property is commonly expressed as a commutative diagramMathworldPlanetmath:


To construct a free Lie algebra is generally and indirect process. We begin with any free associative algebra KX on X, which can be constructed as the tensor algebra over a free K-module with basis X. Then KX- is a K-Lie algebra with the standard commutator bracket [a,b]=ab-ba for a,bKX.

Now define 𝔉𝔏KX as the Lie subalgebra of KX- generated by X.

Theorem 1 (Witt).

[1, Thm V.7] FLKX is a free Lie algebra on X and its universal enveloping algebra is KX.

It is generally not true that 𝔉𝔏KX=KX-. For example, if xX then x2KX but x2 is not in 𝔉𝔏KX.


  • 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