I am using Milne's notes about abelian varieties. There is a proposition that says
For any abelian variety $A$, $$2\dim A \geq [\text{End}^{0}A:\mathbb{Q}]_\text{red}.$$
The proof uses the fact that $\text{End}^{0}(A)$ is semi-simple and it acts faithfully on the $2\dim A$-dimensional vector space $H_{1}(A,\mathbb{Q})$. Then use semi-simplicity to show the rest of the proof.
I got the semi-simpleness and the dimension of $H_{1}$, but not quite sure what is this group action. My naive guess would be: view $A=V/\Lambda$, where $V$ is a vector space of $\dim V=2\dim A$ and $\Lambda$ is a full rank lattice. Then $H_{1}=\Lambda \otimes \mathbb{Q}$. $\text{End}^{0}$ acts on it by acting on the lattice $\Lambda$.
Is this understanding correct? And is there any alternative way to describe this group action?
Thanks!
Yes, that is correct (there is a natural identification of the endomorphisms of $A$ with the endomorphisms of $V$ preserving $\Lambda$).
Alternatively, note that $H_{1}(A,\mathbb{Q})$ is a functor in $A$, and so there is a natural action of $End(A)$ on it. As $H_{1}(A,\mathbb{Q})$ is a $\mathbb{Q}$-vector space, this extends to an action of $End(A)\otimes \mathbb{Q}$.