How to go from Lie Algebra representation to group representation? On wikipedia, they say about $SU(2)$ group : "Since the group SU(2) is simply connected, every representation of its Lie algebra can be integrated to a group representation".
They give a reference to a book but I don't find the corresponding theorem inside.
I would like to basically understand the connection between representations of Lie algebras and Lie groups. I am a huge beginner in representation theory, lie group and lie algebra, I basically know the definitions. 
So I would like a really simple answer (I'm actually doing physics and I need to basically understand the connection between Lie group and Lie algebra representation).
 A: Since you're doing physics, almost every Lie algebra you will encounter can be defined as a Lie algebra of matrices, where the bracket is given by the matrix commutator. In these cases, the elements of the corresponding Lie group can be written as a matrix exponential $g = e^{iX} = \sum_{n = 0}^\infty \frac 1 {n!}(i X)^n$, for some $X$ in the Lie algebra. (I'm assuming the Lie group is connected. And yes, physicists tend to put an $i$ inside the exponential.)
Now, suppose that $X \mapsto \rho(X) \in {\rm M}(n \times n, \mathbb C)$ is a representation of the elements of the Lie algebra as $n \times n$ matrices. Then the corresponding representation of the Lie group is given by the map $ e^{iX} \mapsto e^{i \rho(X)} \in {\rm Gl}(n , \mathbb C)$.
So why is this useful in physics? Suppose you're dealing with a quantum particle of spin $1$. Then the action of the angular momentum operators (generators of the $SU(2)$ Lie algebra) on the wavefunction of your particle are represented by the $3 \times 3$ matrices,
$$ J_x =  \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{array} \right), \ \ \ J_y =  \left( \begin{array}{ccc} 0 & 0 & i \\ 0 & 0 & 0 \\ -i & 0 & 0 \end{array} \right), \ \ \ J_z =  \left( \begin{array}{ccc} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{array} \right) .$$
The transformation of the particle's wavefunction under a spatial rotation about the axis $(n_x, n_y, n_z)$ through an angle $\theta$ is then given by action of the quantum operator $\exp[ i \theta (n_x J_x + n_y J_y + n_z J_z)]$, which is an element of the $SU(2)$ Lie group, and for our spin-one particle, this operator is represented by the matrix
$$\exp[ i \theta (n_x J_x + n_y J_y + n_z J_z) ] =  \exp \left( \begin{array}{ccc} 0 & n_z \theta & -n_y \theta \\ -n_z \theta & 0 &  n_x\theta  \\ n_y \theta & -n_x \theta & 0 \end{array} \right) .$$
There is a similar story for the Lorentz group in relativity, and for the gauge groups in the standard model: in each case, you exponentiate the representation of the Lie algebra to get the representation of the Lie group.
