In the classification of simple lie algebra, I learnt that $\mathfrak{sl}(n + 1)$ has a root system $A_n$. For example http://stacky.net/files/written/LieGroups/LieGroups.pdf, page 82. But I found elsewhere that the simple lie groups corresponding to $A_n$ is $SU(n + 1)$. For example http://en.wikipedia.org/wiki/Simple_Lie_group.
Could somebody explain to me why this is not $SL(n + 1)$ as I would naively expected? Isn't the corresponding simple lie groups are just lie groups generated by the simple lie algebras we just classified?
First of all, different Lie groups may have the same Lie algebra. For example, the Lie algebras of $PSL(n,\mathbb{C})$ and $SL(n,\mathbb{C})$ are both isomorphic to $\mathfrak{sl}_n(\mathbb{C})$. Secondly, different real simple Lie algebras may become isomorphic over the complex numbers, e.g., $\mathfrak{so}_3(\mathbb{R})$ and $\mathfrak{sl}_2(\mathbb{\mathbb{R}})$. The complexification is in both cases isomorphic to $\mathfrak{sl}_2(\mathbb{\mathbb{C}})$.