Direct Sum Decomposition of Semisimple Algebra by Schur's Lemma In Remark $3.1.2$ of Etingof's Introduction to Representation Theory, it says that a semisimple representation $V$ of an algebra $A$ is isomorphic to $\oplus_X Hom_A(X,V)\otimes X$ running over all irreducible representations $X$ of $A$. Specifically, it says this can be deduced from Schur's lemma. Can someone shed some light as to how exactly Schur's Lemma can be used to prove this direct sum decomposition?
 A: Please don't leave out the context when quoting from a book. Not everyone here has the book, and the arXiv version (arXiv:0901.0827v5) has a different numbering (what you call Remark 3.1.2 -- actually it is Remark 3.1.3 in my version of the book! -- is an unlabelled Remark between Definition 2.1 and Proposition 2.2 in the arXiv version).
Part of the missing context is that the ground field $k$ is algebraically closed throughout the book. Another part is that the alleged isomorphism is given by
$\bigoplus\limits_{X \text{ irrep of } A} \operatorname{Hom}_A\left(X, V\right) \otimes X \to V,$
$g \otimes x \mapsto g\left(x\right)$ for all irreps $X$, all $x \in X$ and $g \in \operatorname{Hom}_A\left(X, V\right)$.
It is easy to see that this is an $A$-module map. Why is this an isomorphism? Since $V$ is semisimple, i.e., a direct sum of irreps, we can WLOG assume that $V$ itself is an irrep (because tensor products and $\operatorname{Hom}_A$'s commute with taking direct sums), and then the direct sum
$\bigoplus\limits_{X \text{ irrep of } A} \operatorname{Hom}_A\left(X, V\right) \otimes X$
has only one nonzero addend, namely the one for $X = V$, and this addend is $\operatorname{Hom}_A\left(V, V\right) \otimes V \cong V$ (since Schur's lemma yields $\operatorname{Hom}_A\left(V, V\right) \cong k$).
