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I'm trying to prove that any representation of a semisimple Lie algebra can be uniquely decomposed into irreducible representations.

I have seen some sketches of proofs that show that any representation [tex] \phi [/tex] of a semisimple Lie algebra which acts on a finite-dimensional complex vector space [tex] V [/tex] is completely reducible (i.e. [tex] V = V_1 \oplus V_2 \oplus \cdots \oplus V_k [/tex], such that the restriction of [tex] \phi [/tex] to each [tex] V_i [/tex] is irreducible). But how do we know that this decomposition is unique?

I have seen some sketches of proofs that show that any representation [tex] \phi [/tex] of a semisimple Lie algebra which acts on a finite-dimensional complex vector space [tex] V [/tex] is completely reducible (i.e. [tex] V = V_1 \oplus V_2 \oplus \cdots \oplus V_k [/tex], such that the restriction of [tex] \phi [/tex] to each [tex] V_i [/tex] is irreducible). But how do we know that this decomposition is unique?

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