3
$\begingroup$

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 $X\in su(3)\subseteq sl(3,\mathbb{C})$

  • 0
    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

1 Answers 1

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 $\pi_1(\textrm{SU}(3))=0$.