# Identification of semisimple representations

I'm reading the book Introduction to Representation Theory by Pavel Etingof et all, and I have some questions about the remark $$3.1.3$$. The authors state that by Schur's lemma, any semisimple representation $$V$$ of $$A$$ is caninically identified with $$\bigoplus_{X}Hom_{A}(X, V) \otimes X$$ where $$X$$ runs over all irreducible representations of A. They claim that the function $$f: \bigoplus_{X}Hom_{A}(X, V) \otimes X \to V$$ given by $$g \otimes x \mapsto g(x)$$, $$x \in X$$, $$g \in Hom_{A}(X, V)$$ is an isomorphism. My first question is where is the Schur's lemma being used? And how can i prove that this function is injective? Thank you in advance

All vector spaces are over some fixed field $$k$$, that is often omitted. Let us now write (by definition, non-canonically) $$V$$ as a direct sum of irreducible pieces, $$V=Y_1\oplus Y_2\oplus\dots$$ and use this below \begin{aligned} &\bigoplus_{X}Hom_{A}(X, V) \otimes X \\ &= \bigoplus_{X}Hom_{A}(X,Y_1\oplus Y_2\oplus\dots ) \otimes X \\ &= \bigoplus_{X}\Big(\ Hom_{A}(X,Y_1)\oplus Hom_{A}(X,Y_2)\oplus\dots\ \Big) \otimes X \\ &= \bigoplus_{X}\Big(\ Hom_{A}(X,Y_1)\otimes X\oplus Hom_{A}(X,Y_2)\otimes X\oplus\dots\ \Big) \\ &= \bigoplus_{X}Hom_{A}(X,Y_1)\otimes X\ \oplus\ \bigoplus_{X} Hom_{A}(X,Y_2)\otimes X\oplus\dots \\ &= \bigoplus_{X\ :\ X=Y_1}Hom_{A}(X,Y_1)\otimes X \ \oplus\ \bigoplus_{X\ :\ X=Y_2} Hom_{A}(X,Y_2)\otimes X\ \oplus\ \dots \text{ (Schur)} \\ &= Hom_{A}(Y_1,Y_1)\otimes Y_1 \ \oplus\ Hom_{A}(Y_2,Y_2)\otimes Y_2\ \oplus\ \dots \\ &= k\otimes Y_1 \ \oplus\ k\otimes Y_2\ \oplus\ \dots \text{ (Schur)} \\ &=Y_1\oplus Y_2\oplus\dots \\ &=V\ . \end{aligned} In the chain of isomorphisms above (denoted by abuse with $$=$$) only the first and the last step are non-canonical. So, to see that the declared map (induced on each summand by) $$g\otimes x\to g(x)$$ is canonical, we have only to show that for any $$\text{choice }V\overset f\longrightarrow \underbrace{Y_1\oplus Y_2\oplus\dots}_W$$ we have the commutativity $$\require{AMScd}$$ $$\begin{CD} \bigoplus_{X}Hom_{A}(X, V) \otimes X @>>> \bigoplus_{X}Hom_{A}(X, W) \otimes X \\ @V V V @VV V\\ V @>>f> W \end{CD}$$ thus equivalently for every piece: $$\begin{CD} Hom_{A}(X, V) \otimes X @>>> Hom_{A}(X, W) \otimes X \\ @V V V @VV V\\ V @>>f> W \end{CD}$$ and on elements, the diagram chasing is: $$\begin{CD} g \otimes x @>>> f_*(g) \otimes x \\ @V V V @VV V\\ g(x) @>>f> f(g(x)) \end{CD}$$ which is true.

• I can't see this: $$\bigoplus_{X} Hom_{A}(X, Y_{1}) \otimes X \oplus ... \oplus \bigoplus_{X} Hom_{A}(X, Y_{n}) \otimes X \cong$$ $$\bigoplus_{X: X \cong Y_{1}}Hom_{A}(X, Y_{1}) \otimes X \oplus ... \oplus \bigoplus_{X: X \cong Y_{n}} Hom_{A}(X, Y_{n}) \otimes X$$ – Jaime Grimal Alves Sep 8 at 23:26
• The theorem of Schur, en.wikipedia.org/wiki/…, tells us that the set of homomorfisms between $X$ and $Y_1$, both irreducible, is $0$, when they are not isomorphic. So we can restrict to the case of the only one $X$ in the list under the big direct sum of all irreducible representations which is $\cong Y_1$. – dan_fulea Sep 9 at 1:04
• Can these isomorphisms be seen as A-isomorphisms, that is, module isomorphisms? – Jaime Grimal Alves Sep 19 at 15:41

Schur's lemma says, some map under some conditions is an isomorphism.

Your question is to see the obvious map $$\bigoplus_X (\text{Hom}_A(X,V)\otimes X)\rightarrow V$$ to be an isomorphism.

Suppose for simplicity $$V$$ is an irreducible representation of $$A$$.

Hint : How many $$A$$-linear maps can you think of from $$X$$ to $$V$$ if $$X$$ and $$V$$ are not isomorphic?

As $$V$$ is semisimple, it is direct sum of irreducible representations. Suppose $$V=V_1\oplus V_2$$.

Check : Does the following equality holds? $$\bigoplus_X (\text{Hom}_A(X,V)\otimes X)=\bigg( \bigoplus_X (\text{Hom}_A(X,V_1)\otimes X)\bigg)\bigoplus \bigg( \bigoplus_X (\text{Hom}_A(X,V_2)\otimes X)\bigg)$$ Now, $$X$$ and $$V_1,V_2$$ are irreducible representations. Can you see the result?