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I read something about Lie algebra where it requires a representation $\rho : \mathfrak g \to \mathfrak{gl}(V)$ to be "semisimple and irreducible". In my understanding, a representation is semisimple just means it is completely reducible, i.e. it is the direct sum of some irreducible representations. Hence if it is irreducible, it is automatically semisimple. Is it right?

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Yes, you are right in every aspect. Therefore, “semisimple and irreducible” is the same thing as “irreducible”.

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  • $\begingroup$ Thanks. I just don't understand why the author does not simply says irreducible. $\endgroup$ – Akatsuki Jun 5 '18 at 17:17
  • $\begingroup$ @Akatsuki I have no idea. Can you provide a link to the text? $\endgroup$ – José Carlos Santos Jun 5 '18 at 17:19
  • $\begingroup$ Yes, see here. It's on page 3, line 4 of section 1. $\endgroup$ – Akatsuki Jun 5 '18 at 17:28
  • $\begingroup$ @Akatsuki It's even stranger than what you thought. It's a finite-dimensional represention of a simple Lie algebra over $\mathbb C$. Every such representation is semisimple! $\endgroup$ – José Carlos Santos Jun 5 '18 at 17:31
  • $\begingroup$ I didn't see where it says $\mathfrak g$ is simple? $\endgroup$ – Akatsuki Jun 5 '18 at 17:35

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