  We extend significantly previous works on the Hilbert space representations
of the Generalized Uncertainty Principle (GUP) in 3+1 dimensions of the form
$[X_i,P_j] = i F_{ij}$ where $ F_{ij} = f(P^2) \delta_{ij} + g(P^2) P_i P_j $
for any functions $f$. However, we restrict our study to the case of commuting
$X$'s. We focus in particular on the symmetries of the theory, and the minimal
length that emerge in some cases. We first show that, at the algebraic level,
there exists an unambiguous mapping between the GUP with a deformed quantum
algebra and a quadratic Hamiltonian into a standard, Heisenberg algebra of
operators and an aquadratic Hamiltonian, provided the boost sector of the
symmetries is modified accordingly. The theory can also be mapped to a
completely standard Quantum Mechanics with standard symmetries, but with
momentum dependent position operators. Next, we investigate the Hilbert space
representations of these algebraically equivalent models, and focus,
specifically on whether they exhibit a minimal length. We carry the functional
analysis of the various operators involved, and show that the appearance of a
minimal length critically depends on the relationship between the generators of
translations and the physical momenta. In particular, because this relationship
is preserved by the algebraic mapping presented in this paper, when a minimal
length is present in the standard GUP, it is also present in the corresponding
Aquadratic Hamiltonian formulation, despite the perfectly standard algebra of
this model. In general, a minimal length requires bounded generators of
translations, i.e. a specific kind of quantization of space, and this depends
on the precise shape of the function $f$ defined previously. This result
provides an elegant and unambiguous classification of which universal quantum
gravity corrections lead to the emergence of a minimal length.
