Friedrichs’ theorem

Fix a commutative unital ring $K$ of characteristic 0. Let $X$ be a finiteset and $K⟨X⟩$ the free associative algebra on $X$. Then definethe 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 universalenveloping algebra of a Lie algebra. Here the Lie algebra in question isthe free Lie algebra on $X$, $FL⟨X⟩$ whose universal envelopingalgebra is $K⟨X⟩$ by a theorem of Witt.

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