Which are the underlying Lie group and algebra related to the translation invariance in field theories? I'm new to Physics SE. I've seen a lot of interesting questions and answers, and thought it will be very useful to participate a little. 
I'm currently stuck in a, probably, very simple matter, regarding the nature of the linear momentum $P_{\alpha}$ in field theory. I know it is commonly known as the "infinitesimal generator of translations", pointing an obvious relation with the Lie algebra generators. But which are the Lie group, and correspondingly Lie algebra, associated with $P_{\alpha}$?
I thought of something like the following: if $M$ is the Minkowski spacetime, let $G = (M,+)$ be the Lie group, therefore $G$ acts on $M$. So if $x^{\alpha}\in M$ and $g\in G$, then:
$g(x^{\alpha}) = x^{\alpha} + \delta^{\alpha}$,
but I've been struggling to mathematically write the relation between the algebra generators, the exponential map and the momentum operator $P^{\alpha}$. 
Does anyone knows how to point me in the right direction? Thank you all!
 A: The group is $\mathbb{R}^4$ (tuples of 4 real numbers), group product is vector space addition, group inverse is the opposite vector, and group identity is the zero vector. Easy to see that the group is Abelian (commutative).
Corresponding Lie algebra is $4 \mathfrak{u}_1$, which also consists of tuples of 4 numbers, with the Lie bracket identically zero for any two elements.
This is reflected in the relation
$$ [ P_{\mu}, P_{\nu} ] = 0. $$
It follows that 4-momenta components are simultaneously diagonalizable, which enables us to speak about the spectrum of the quantum field theory, and also that the momentum conservation law holds (as the components of 4-momenta commute with the Hamiltonian $P_0$).
$\mathbb{R}^4$ is the maximally noncompact form of $4 \mathfrak{u}_1$, the other forms having one or more of their dimensions compactified on a circle. Of course in reality space is infinitely extended... probably. Or not.
As mentioned in the comments, this is only part of the Poincare group/algebra, which also contains rotations and boosts (Lorentz transformations of the reference frame).
