$V$ is a finite dimensional module for $\mathfrak{sl}(2,\mathbb{C})$. Show $V$ is determined up to isomorphism by the eigenvalues of $h$. Suppose that $V$ is a finite dimensional module for $\mathfrak{sl}(2,\mathbb{C})$. I'm trying to use Weyl's Theorem and the the well known classification of irreducible modules for $\mathfrak{sl}(2,\mathbb{C})$, that $V$ is determined up to isomorphism by the eigenvalues of $h$ ($h$ is the matrix with $1$ in the upper left, $-1$ in the lower right, and 0 in the other entries).
In Particular, I want to show that if $V$ is a direct sum of $k$ irreducible modules, then $k = \dim(W_0) + \dim(W_1)$ where $W_r=\{v \in V: h \cdot v = rv\}$
Can someone show me what's going on here? I'm just a math hobbyist and have been stuck on this for awhile. I'd appreciate if someone could show me what's going on! I'm sick of trying to reinvent the wheel on this one.
 A: $h\in\mathfrak{sl}(2,\Bbb{C})$ is a semisimple element (meaning that it is diagonal) and by preservation of the Jordan decomposition, it acts diagonally on $\mathfrak{sl}(2,\Bbb{C})-$representations. This is how we produce the classification of irreducible $\mathfrak{sl}(2,\Bbb{C})-$representations. From here on I'll just write $\mathfrak{g}=\mathfrak{sl}(2,\Bbb{C})$.
Let $V$ be the module of highest weight $\lambda$, meaning that the highest eigenvalue of $h$ is $\lambda \in \Bbb{Z}_{\ge 0}$. The integrality of the weight is a basic theorem. We use this to classify $\mathfrak{g}-$representations by their highest weights. Such a $V$ can be written as
$$ V=\bigoplus_{\mu=0}^\lambda V_{\lambda-2\mu} $$
where $V_{\tau}=\{x\in V:hx=\tau x\}$ are the weight spaces (eigenspaces in this case).
Now, if we have a finite dimensional $\mathfrak{g}-$module $W$, we can use Weyl's Theorem on complete irreducibility to write it as $W=\bigoplus_{i=1}^r W_i$ where the $W_i$ are irreducible. If $W_i$ corresponds to an odd highest weight $\lambda_i$, then among the list of weights $\lambda_i,\lambda_i-2,\ldots, -\lambda_i$ is $1$, and conversely. If $W_j$ has highest weight $\lambda_i$, which is even, then $\lambda_j-\frac{\lambda_j}{2}2=0$ is a weight appearing in this module. Since weights are either odd or even, it suffices to count the multiplicity of the weights $0$ and $1$ in $W$. This is the same thing as counting the multiplicities of the eigenvalues $0$ and $1$ for $h$ acting on $W$. This gives
$$ r=\dim W_0+\dim W_1.$$
