One variant of Schur's Lemma states that $$ \text{Hom}(S,T) \cong \left\{ \begin{matrix} 0 & \text{if } S \neq T \\ \mathbb{C} & \text{if } S = T \end{matrix} \right. $$ when $S,T$ are simple modules over a $\mathbb{C}$-algebra $A$.

I would like a similar result for $\text{Hom}(\bigoplus_{i=1}^n S_i,S)$. Will it be isomorphic to $ \bigoplus_{i=1}^n \text{Hom}(S_i,S) $ or is this just wishful thinking?

EDIT: and the same question for $ \text{Hom}(\bigoplus S_i,\bigoplus T_j) $

EDIT 2: OK i know where i was confused, for some reason i thought $\text{Hom}(U,V)$ was an algebra when it's actually a module

  • 1
    $\begingroup$ For your deleted question: $\Bbb C\oplus\Bbb C$ and $\Bbb C[\varepsilon]/(\varepsilon^2)$ are nonisomorphic two-dimensional $\Bbb C$-algebras. | Yes $\hom$ distributes over (arbitrary) direct sums in both arguments. In fact there is a canonical isomorphism. Can you tell what it is? $\endgroup$ – anon Dec 27 '14 at 21:32
  • $\begingroup$ I'm good now, thanks for all the help! $\endgroup$ – JC574 Dec 27 '14 at 21:45
  • 2
    $\begingroup$ @JC574 For your deleted question (I think you should undelete it), the answer is yes if the algebras are commutative and semisimple. $\endgroup$ – egreg Dec 27 '14 at 21:52
  • $\begingroup$ @egreg If I get time later tonight I'll reask it and tag you. Thanks for your help. $\endgroup$ – JC574 Dec 27 '14 at 22:14

If $S, S_i, T, T_i$ are simple modules, then $$ \dim\text{Hom}(\bigoplus_{i=1}^n S_i,S) = \sum_{i=1}^n \dim \text{Hom} (S_i, S). $$ is the number of $S_i$ isomorphic to $S$.

For $$ \dim \text{Hom}(\bigoplus S_i,\bigoplus T_j) $$ you get the "same thing".

  • $\begingroup$ so we have dimensions, but no algebra isomorphism necessarily? $\endgroup$ – JC574 Dec 27 '14 at 20:17
  • 1
    $\begingroup$ @JC574: Sorry, yes, we have isomorphisms. I just tend to think in terms of dimensions. You just get copies of $\mathbb{C}$. $\endgroup$ – Thomas Dec 27 '14 at 20:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.