# Semisimple Representation and Irreducible Representation

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?

• @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! – José Carlos Santos Jun 5 '18 at 17:31
• I didn't see where it says $\mathfrak g$ is simple? – Akatsuki Jun 5 '18 at 17:35