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The Killing Form Takes a Trace of Two Adjoint Maps by Vicfred

tags: mathematics, Lie algebras, brackets, matrix groups

The Killing form is an invariant symmetric bilinear form that detects semisimplicity in characteristic zero. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Objects and notation

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

$$ B(x,y)=\operatorname{tr}(\operatorname{ad}_x\operatorname{ad}_y) $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ B([z,x],y)+B(x,[z,y])=0 $$

Push the symbols

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ \left(B_{\mathfrak{sl}_2}(x_i,x_j)\right)_{e,f,h}=\begin{pmatrix}0&4&0\\4&0&0\\0&0&8\end{pmatrix} $$

Structural reading

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.

$$ \begin{aligned} \mathsf{D}\;&:\quad B(x,y)=\operatorname{tr}(\operatorname{ad}_x\operatorname{ad}_y),\\[5pt] \mathsf{C}\;&:\quad B([z,x],y)+B(x,[z,y])=0. \end{aligned} $$

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.

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

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 Fri 12 January 2024. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.