I was reading about representation of lie algebra;.We know, If a representation is irreducible that implies completely reducible. What about converse. does completely reducible implies irreducible? any examples
2026-04-09 17:57:34.1775757454
Does completely reducible implies irreducible
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By definition, the direct sum of two irreducible representations of a Lie algebra $L$ is completely reducible, but not irreducible. Note that Weyl’s Theorem says that any finite dimensional representation of a semisimple Lie algebra $L$ over a field of characteristic zero is completely reducible.