Knowlpedia

Example:

The 3D simple Lie algebra of traceless complex matrices with standard relations.

Let be the Lie algebra of traceless 2×22\times 2 complex matrices with bracket the commutator.

Standard basis and brackets (explicit calculation)

Define

E=(0100),F=(0010),H=(100−1).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}.

Compute:

[H,E]=HE−EH=(0200)=2E,[H,F]=HF−FH=(00−20)=−2F,[H,E]=HE-EH =\begin{pmatrix}0&2\\0&0\end{pmatrix}=2E,\qquad [H,F]=HF-FH =\begin{pmatrix}0&0\\-2&0\end{pmatrix}=-2F,

and

[E,F]=EF−FE=(100−1)=H.[E,F]=EF-FE =\begin{pmatrix}1&0\\0&-1\end{pmatrix}=H.

These relations imply [sl2,sl2]=sl2[\mathfrak{sl}_2,\mathfrak{sl}_2]=\mathfrak{sl}_2, so it is perfect (see ) and in fact .

Cartan and root decomposition

A is h=CH\mathfrak h=\mathbb C H. The adjoint action adH\mathrm{ad}_H is diagonalizable on sl2\mathfrak{sl}_2 with

adH(E)=2E,adH(F)=−2F,adH(H)=0.\mathrm{ad}_H(E)=2E,\qquad \mathrm{ad}_H(F)=-2F,\qquad \mathrm{ad}_H(H)=0.

Thus the has one-dimensional root spaces:

g2=CE,g−2=CF,g0=CH.\mathfrak g_{2}=\mathbb C E,\quad \mathfrak g_{-2}=\mathbb C F,\quad \mathfrak g_{0}=\mathbb C H.

With an appropriate positivity choice, 22 is the positive root and −2-2 the negative root (compare ).

Killing form (quick computation pattern)

The is nondegenerate, consistent with . For example, one can compute κ(H,H)=8\kappa(H,H)=8 and κ(E,F)=4\kappa(E,F)=4 by evaluating traces of ad\mathrm{ad}-operators in this basis.

Context. sl2(C)\mathfrak{sl}_2(\mathbb C) is the local model for rank-1 semisimple theory; many general results (weights, highest-weight modules) can be tested concretely here (see ).