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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." –  Qiaochu Yuan Oct 9 '12 at 8:37
    
And the phrase requires plurals: "representations" (twice) since a ono-to-one correspondence between singlton sets is not very exciting. –  Marc van Leeuwen Oct 9 '12 at 9:12
    
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 $sl(3,\Bbb C)\to gl(V)$. –  Marc van Leeuwen Oct 9 '12 at 9:18

1 Answer 1

Any complex representation $\Pi:\textrm{SU}(3) \to \textrm{GL}(V)$ determines a unique complex representation $\pi:\mathfrak{sl}_3 \to \mathfrak{gl}(V)$ such that $$\Pi(e^{X}) = e^{\pi(X)}$$ for all $X$ in the Lie algebra. That any complex representation of $\mathfrak{sl}_3$ on $\mathfrak{gl}(V)$ determines a unique Lie group representation of $\textrm{SU}(3)$ on $\textrm{GL}(V)$ with the property above comes from the fact that $\pi_1(\textrm{SU}(3))=0$.

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