Source:
Lecture Notes in Lie Algebra by Kailash C. Misra
Let $L= sl(2,\mathcal{C})$ and $V=gl(2,\mathcal{C})$. Then V is an L-module via the action $x\cdot v = xv$(matrix multiplication) for all $x \in L$, $v \in V$. So by Weyl's Theorem V is completely reducible. Find irreducible L-submodules $W_1$ and $W_2$ such that $V=W_1\bigoplus W_2$.
I'm truly at a loss here as to where to start.
I can't think of what L-submodules look like.
I can't think of a unique (if indeed it need be unique) submodule decomposition of V.
- The only thing that comes to mind is something from topology, where you describe the real plane as a direct sum of the upper half plane and the lower half plane. Only intuition at this point :/
Hint: $gl(2, \Bbb C)$ here is really just a confusing way to write the $2\times 2$-matrices with entries in $\Bbb C$. Now a standard simple module on which $sl(2, \Bbb C)$ acts by multiplication from the left is the column vectors $\pmatrix{x\\y}$. Can you find such columns in the $2\times2$-matrices?
Added: To spell it out completely, look at
$$V_1 := \lbrace \pmatrix{x&0\\y&0}: x,y \in \Bbb C \rbrace$$
$$V_2 := \lbrace \pmatrix{0&w\\0&z}: w,z \in \Bbb C \rbrace$$
Show that both are $L$-submodules under the given operation (i.e. show that they are subspaces of $V$, and $L \cdot V_i \subseteq V_i$), and that $V \simeq V_1 \oplus V_2$ as vector spaces.