Ray representations and Galilei group

For my PhD thesis I performed a work in group teory, precisely in the theory of representations, applied to quantum mechanics. So, in order to describe my work, recently published by the Journal of Mathematical Physics, I need to introduce some concepts. The group theory was founded indipendetly by Niels Abel and Evariste Galois and it is focused on group, a set $G$ of elements with a multiplication operation $\cdot$ and such that the following properties are true⁽⁹⁾:

1. $\forall a, b \in G, a \cdot b \in G$
2. $\forall a, b, c \in G, a(bc) = (ab) c$
3. $\forall a \in G \, \exists e \in G \, \text{:} \, ae = ea = a$
4. $\forall a \in G \, \exists b \in G \, \text{:} \, ab = ba = e$ and $b = a^{-1}$

A group is called abelian group if

1. $\forall a, b \in G, ab = ba$

For every group, we can write a representation, that is a set of operators or functions that act in a mathematical space and change it in a same way at the elements of the group change their own space⁽¹⁾. In other (trivial) words we have a given world with a group (for example our real world with a group of symmetry, for example the rotations), and in order to find properties of this world we must use a mathematical representation.
If the world is a given physical system (for example a free particle), we have a symmetry group, that is a set of all symmetry transformation⁽²⁾ of our physical system, and his representation acts in a so called Hilbert space. In this space, following Wigner's theorem⁽⁴⁾, the most general representation is a ray (unitary) representation. In order to understand the ray (or projective) representations, we must enunciate the theorem:

For every transformation of symmetry $T: \mathcal R \rightarrow \mathcal R$ between the rays of a Hilbert space $\mathcal H$ and such that conserve the transition probabilities, we can define an operator $U$ on the Hilbert space $\mathcal H$ such that, if $|\psi> \in {\mathcal R}_\psi$, then $U |\psi> \in {\mathcal R}'_\psi$, where ${\mathcal R}_\psi$ is the radius of the state $|\psi>$, ${\mathcal R}'_\psi = T {\mathcal R}_\psi$, and $U$ uniform and linear \[< U \psi | U \varphi> = <\psi | \varphi>, \qquad U |\alpha \psi + \beta \varphi> = \alpha U |\psi> + \beta U |\varphi>\] or with $U$ antiunitario and antilinear: \[< U \psi | U \varphi> = <\varphi | \psi>, \qquad U |\alpha \psi + \beta \varphi> = \alpha^* U |\psi> + \beta^* U |\varphi>\] Further, $U$ is uniquely determined except for a phase factor.

So a ray representation is the association between an element of the symmetry group $G$ to a set of unitary (or antiunitary) operators which differ only for a phase: in other worlds a ray of operators⁽³⁾.