# Friedrichs’ theorem

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

###### Theorem 1 (Friedrichs).

[1, Thm V.9] An element $a\in K\langle X\rangle$ is a Lie element if and only if $a\delta=a\otimes 1+1\otimes a$.

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 $X$, $FL\langle X\rangle$ whose universal enveloping algebra is $K\langle X\rangle$ by a theorem of Witt.

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

## References

• 1 Nathan Jacobson Lie Algebras, Interscience Publishers, New York, 1962.
Title Friedrichs’ theorem FriedrichsTheorem 2013-03-22 16:51:16 2013-03-22 16:51:16 Algeboy (12884) Algeboy (12884) 6 Algeboy (12884) Theorem msc 16S30 msc 17B35