I am new to concepts from Lie algebra theory, so I was reading Lectures on Lie groups ad Lie algebras by Carter, Macdonald and Segal.
At the third chapter we are introducing universal enveloping algebra of any Lie algebra, there it is claimed $U(\mathfrak{g})$ has the same representation theory as $\mathfrak{g}$ :
but I couldn't understand what does it mean they have "same" representation theory.
I think what he explained is that any representation $\rho$ of $\mathfrak{g}$ induces a representation of $\rho'$ of $U(\mathfrak{g}$) and vice versa, but I am not sure if it is meant that map $\rho \rightarrow \rho'$ is bijective?
The reason for asking about bijectivity is similar to: for every odd integer, there is a even integer (say by ading $1$), and also for every even integer there is a odd integer (say also by adding $1$). But one probably wants to hear we subtract $1$ from even integers to make the map bijective.