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I would like to understand the following fact, shall need help, Thank you.

" There is a one- to- one correspondence between the finite dimensional complex representation $\Pi$ of $SU(3)$ and finite dimensional complex linear representation $\pi$ of $sl(3,\mathbb{C})$ and the correspondence is determined by the property that $$\Pi(e^X)=e^{\pi(X)}$$ for all $X\in su(3)\subseteq sl(3,\mathbb{C})$

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    The phrase is "one-to-one correspondence."2012-10-09
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    And the phrase requires plurals: "representations" (twice) since a ono-to-one correspondence between singlton sets is not very exciting.2012-10-09
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    Or maybe, on second reading, the singular is intended, but that makes the intention rather murky: I think it says that for a given finite dimensional complex vector space $V$ there is a correspondence between the sets of Lie group morphisms $SU(3)\to GL(V)$ and of Lie algebra morphisms $sl(3,\Bbb C)\to gl(V)$.2012-10-09

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