Lagmental Vicfred

The Jacobi Identity Is a Cyclic Cancellation Law by Vicfred

tags: mathematics, Lie algebras, brackets, matrix groups

The defining three-term identity of a Lie algebra makes nested brackets compatible with derivations. This is a compact note, but the quantifiers and hypotheses stay on the page.

Definitions first

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

$$ [x,y]=-[y,x] $$

I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).

$$ [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0 $$

A small case in full

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ \underbrace{[X,[Y,Z]]}_{XYZ-XZY-YZX+ZYX}+\underbrace{[Y,[Z,X]]}_{YZX-YXZ-ZXY+XZY}+\underbrace{[Z,[X,Y]]}_{ZXY-ZYX-XYZ+YXZ}=0 $$

The reusable statement

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad [x,y]=-[y,x],\\[5pt] \mathsf{C}\;&:\quad [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0. \end{aligned} $$

A nearby false statement

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0 \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Mon 17 July 2023. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.