I'm interested in seeing the computation for the fundamental groups of the simple algebraic groups of type $A$. Below is the definition of the fundamental group for a simple algebraic group $G$.
Let $G$ be a simple algebraic group over a field $k$, and fix a maximal torus $T\subset G$. The adjoint action of $T$ on $\mathfrak{g}=\mathrm{Lie}(G)$ gives a root space decomposition of $\mathfrak{g}$ which corresponds to an abstract root system $\Phi$ inside of $E=\mathbb{R}\otimes_{\mathbb{Z}}X(T)$. Here $X(T)$ is the free abelian group of characters of $T$.
Inside of $E$ there are abstract weights which are defined to be all $\lambda\in E$ such that $\langle \lambda,\alpha\rangle\in\mathbb{Z}$ for all $\alpha\in\Phi$, where
$$\langle\lambda,\alpha\rangle=\frac{2(\lambda,\alpha)}{(\alpha,\alpha)}$$
and $(\cdot,\cdot)$ is the (unique?) inner-product on $E$ relative to which the Weyl group $W(\Phi)$ consists of orthogonal transformations. Let $\Lambda$ be the lattice of abstract weights, and $\Lambda_r$ the lattice generated by $\Phi$. We have the following inclusion of lattices:
$$\Lambda_r\subset X(T)\subset \Lambda$$
The fundamental group of $G$ is $\pi(G)=\Lambda/X(T)$. We say that $G$ is simply-connected if $\pi(G)=0$ and $G$ is adjoint if $X(T)=\Lambda_r$.
In the case of type $A$ groups, I've read that the simply-connected group is $SL_n(k)$, and the adjoint group is $PGL_n(k)$. I'd very much appreciate it is someone could describe how to arrive at the three lattices described above for each of these groups, to see that this claim is in fact true. I'd also be interested in seeing the other groups of type $A$ which correspond to the intermediate lattices $X(T)$. References are appreciated too. Thanks!
Edit: If people are avoiding this question because it asks too much, I would also be happy with a brief outline of the computation, a few hints, or even just the first step (finding the root decomposition of $\mathfrak{sl}_n(k)$ from the adjoint action of diagonal matrices of determinant $1$).