Direct sum decomposition of sl(2,C)

Question

Source:

Lecture Notes in Lie Algebra by Kailash C. Misra

a)

Let $\phi: M \rightarrow sl(2,\mathcal{C})$ be defined by

$E_{11}-E_{22} \mapsto h$

$E_{12} \mapsto e$

$E_{21} \mapsto f$

L is an M-Module under the adjoint action since the adjoint action simply turns the L-Module conditions in the Lie algebra conditions.

b) Here is where I'm struggling. Any hints?

• I can't think of what M-submodules look like.
• I can't think of a unique (if indeed it need be unique) submodule decomposition of L.

Answer 1

My adviser, peer, and I figured it out, I'll let the reader verify.

$M = span\{E_{11}-E_{22}, E_{12}, E_{21}\}$

$M' = span\{E_{23}, E_{13}\}$

$M'' = span\{E_{31}, E_{32}\}$

$M''' = span\{\frac{1}{2}E_{11}+\frac{1}{2}E_{22}-E_{33}\}$

$L=M\bigoplus M'\bigoplus M'' \bigoplus M'''$