Lagmental Vicfred

The Adjoint Representation Is Bracketing by an Element by Vicfred

tags: mathematics, Lie algebras, brackets, matrix groups

Every Lie algebra acts on itself through the adjoint maps ad x. I want the notation, the mechanism, and the failure mode visible at the same time.

Start locally

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

$$ \operatorname{ad}_x:\mathfrak g\to\mathfrak g,\qquad\operatorname{ad}_x(y)=[x,y] $$

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 \(\Longleftrightarrow\).

$$ [\operatorname{ad}_x,\operatorname{ad}_y]=\operatorname{ad}_{[x,y]} $$

Compute before generalising

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ \begin{aligned}[\operatorname{ad}_x,\operatorname{ad}_y](z)&=[x,[y,z]]-[y,[x,z]]\\&=[[x,y],z].\end{aligned} $$

The global view

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad \operatorname{ad}_x:\mathfrak g\to\mathfrak g,\qquad\operatorname{ad}_x(y)=[x,y],\\[5pt] \mathsf{C}\;&:\quad [\operatorname{ad}_x,\operatorname{ad}_y]=\operatorname{ad}_{[x,y]}. \end{aligned} $$

Edge conditions

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] [\operatorname{ad}_x,\operatorname{ad}_y]=\operatorname{ad}_{[x,y]} \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Wed 27 October 2021. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.