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?
2026-03-26 17:51:56.1774547516
Semisimple Representation and Irreducible Representation
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Yes, you are right in every aspect. Therefore, “semisimple and irreducible” is the same thing as “irreducible”.