A quesion in Fulton & Harris book "representation theory a first course"

Question

In Section 11.2 A little plethysm, it discusses the tensor product of two different representations of $sl_2\mathbb{C}$. It says

"If $V=\bigoplus V_{\alpha}$ and $W=\bigoplus W_{\beta}$ then $V\otimes W=\bigoplus(V_{\alpha}\times W_{\beta})$ and $V_{\alpha}\otimes W_{\beta}$ is an eigenspace for $H$ with eigenvalue $\alpha+\beta$."

Here $V$ and $W$ are irreducible representations of $sl_2\mathbb{C}$ and $H=\left(\begin{array}{cc}1 & 0\\ 0 & -1\end{array}\right)$ is one of the basis element in $sl_2\mathbb{C}$.

My question is the last sentence. Why is $V_{\alpha}\otimes W_{\beta}$ an eigenpace for $H$ with eigenvalue $\alpha+\beta$ not $\alpha\cdot\beta$?

Thanks.

Answer 1

Recall how elements of a lie algebra act on pure tensors:

$$g(v\otimes w)=(gv)\otimes w+v\otimes (gw).$$

This is different from the idea of group action, where $g(v\otimes w)=(gv)\otimes(gw)$.