Lagmental Vicfred

Baker--Campbell--Hausdorff Corrects Noncommutative Addition by Vicfred

Last updated: Sun 15 March 2026

tags: mathematics, Lie algebras, brackets, matrix groups

The logarithm of exp X exp Y begins with X plus Y and then adds nested commutators. I will separate the object being defined from the consequence being claimed.

Objects and notation

A Lie algebra replaces multiplication by a bilinear bracket \([x,y]\) satisfying antisymmetry and Jacobi. Matrix Lie algebras use the commutator \([X,Y]=XY-YX\).

$$ Z=\log(e^Xe^Y) $$

A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.

$$ Z=X+Y+\frac12[X,Y]+\frac1{12}[X,[X,Y]]+\frac1{12}[Y,[Y,X]]+\cdots $$

Push the symbols

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ \left.[X,Y]=0\ \Longrightarrow\ \log(e^Xe^Y)=X+Y\right.,\qquad e^Xe^Y=e^{X+Y} $$

Structural reading

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad Z=\log(e^Xe^Y),\\[5pt] \mathsf{C}\;&:\quad Z=X+Y+\frac12[X,Y]+\frac1{12}[X,[X,Y]]+\frac1{12}[Y,[Y,X]]+\cdots. \end{aligned} $$

A hypothesis worth keeping

The bracket is not associative multiplication. The Jacobi identity controls its failure to associate and makes adjoint maps into a representation.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] Z=X+Y+\frac12[X,Y]+\frac1{12}[X,[X,Y]]+\frac1{12}[Y,[Y,X]]+\cdots \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Thu 21 November 2024. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.