Lagmental Vicfred

The sl2 Commutation Relations Fit in Three Generators by Vicfred

tags: mathematics, Lie algebras, brackets, matrix groups

The matrices e, f, and h generate sl2 and encode its root decomposition. I want the notation, the mechanism, and the failure mode visible at the same time.

The data

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

$$ e=\begin{pmatrix}0&1\\0&0\end{pmatrix},\quad f=\begin{pmatrix}0&0\\1&0\end{pmatrix},\quad h=\begin{pmatrix}1&0\\0&-1\end{pmatrix} $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ [h,e]=2e,\qquad[h,f]=-2f,\qquad[e,f]=h $$

Derivation

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.

$$ \begin{array}{c|ccc}[\ ,\ ]&e&f&h\\\hline e&0&h&-2e\\f&-h&0&2f\\h&2e&-2f&0\end{array} $$

Invariant content

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad e=\begin{pmatrix}0&1\\0&0\end{pmatrix},\quad f=\begin{pmatrix}0&0\\1&0\end{pmatrix},\quad h=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\\[5pt] \mathsf{C}\;&:\quad [h,e]=2e,\qquad[h,f]=-2f,\qquad[e,f]=h. \end{aligned} $$

Scope

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] [h,e]=2e,\qquad[h,f]=-2f,\qquad[e,f]=h \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Thu 12 March 2026. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.