Friedrichs’ theorem

Fix a commutativePlanetmathPlanetmathPlanetmath unital ring K of characteristicPlanetmathPlanetmath 0. Let X be a finite set and KX the free associative algebra on X. Then define the map δ:KXKXKX by xx1+1x.

Theorem 1 (Friedrichs).

[1, Thm V.9] An element aKX is a Lie element if and only if aδ=a1+1a.

The term Lie element applies only when an element is taken from the universal enveloping algebra of a Lie algebraMathworldPlanetmath. Here the Lie algebra in question is the free Lie algebra on X, FLX whose universal enveloping algebra is KX by a theorem of Witt.

This characterization of Lie elements is a primary means in modern proofs of the Baker-Campbell-Hausdorff formula.


  • 1 Nathan Jacobson Lie Algebras, Interscience Publishers, New York, 1962.
Title Friedrichs’ theorem
Canonical name FriedrichsTheorem
Date of creation 2013-03-22 16:51:16
Last modified on 2013-03-22 16:51:16
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 6
Author Algeboy (12884)
Entry type Theorem
Classification msc 16S30
Classification msc 17B35