Friedrichs’ theorem

Fix a commutative unital ring $K$ of characteristic 0. Let $X$ be a finite set and $K⟨X⟩$ the free associative algebra on $X$. Then define the map $\delta :K⟨X⟩\to K⟨X⟩\otimes K⟨X⟩$ by $x\mapsto x\otimes 1+1\otimes x$.

Theorem 1 (Friedrichs).

[1, Thm V.9] An element $a\mathrm{\in }K\mathit{}\mathrm{⟨}X\mathrm{⟩}$ is a Lie element if and only if $a\mathit{}\delta \mathrm{=}a\mathrm{\otimes }\mathrm{1}\mathrm{+}\mathrm{1}\mathrm{\otimes }a$.

The term Lie element applies only when an element is taken from the universal enveloping algebra of a Lie algebra. Here the Lie algebra in question is the free Lie algebra on $X$, $FL⟨X⟩$ whose universal enveloping algebra is $K⟨X⟩$ by a theorem of Witt.

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

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