Physics notation of infinitesimal Lie group Given a Lie group $G$  and given $g \in G$ in most physics books we have this notation.
$$g=I+\epsilon A$$
where $I$ is the identity and $A$ is a generator of the Lie group.
My question what this notation means, since first, we do not have a "+" operation  in the Lie group and second that the generator $A$ and the identity live in different spaces?
 A: You are correct that a Lie Group does not have a priori an additive operation. However, often, we can represent the Lie group as a group of matrices, in which the operation is understood to be Matrix addition. In physics, we always work from the understanding that really we are working with a Matrix Lie group. 
You are correct also in saying that the Generator $A$ lives in a different space. In fact, it it lives in the tangent space of the Lie group when we understand the Lie group as a manifold. Moreover, the generators form something called a Lie algebra. In physics, we more often than not work with the Lie Algebra rather than the Lie Group.
With regards to your notation, we consider the Lie group structure around its identity element. Suppose there is some curve $\gamma:(-\infty,\infty) \rightarrow G$ in the Lie group sending $t\mapsto\gamma(t)\in G$. Suppose $\gamma(0)=I$ is the identity of the Lie group. As a first order expansion around $\epsilon\approx 0$, we write that $$\gamma(\epsilon)=\gamma(0)+\frac{d\gamma(t)}{dt}\Big|_{t=0}\epsilon+\mathcal{O}(\epsilon^2).$$ We thus identify the generator $\frac{d\gamma(t)}{dt}\Big|_{t=0}=A$, being the generator of the one parameter group of Lie Group $\gamma(t)$.
