  Recently, it was shown that when reference frames are associated to quantum
systems, the transformation laws between such quantum reference frames need to
be modified to take into account the quantum and dynamical features of the
reference frames. This led to a relational description of the phase space
variables of the quantum system of which the quantum reference frames are part.
While such transformations were shown to be symmetries of the system's
Hamiltonian, the question remained unanswered as to whether they enjoy a group
structure, similar to that of the Galilei group relating classical reference
frames in quantum mechanics. In this work, we identify the canonical
transformations on the phase space of the quantum systems comprising the
quantum reference frames, and show that these transformations close a group
structure defined by a Lie algebra, which is different from the usual Galilei
algebra of quantum mechanics. We further find that the elements of this new
algebra are in fact the building blocks of the quantum reference frames
transformations previously identified, which we recover. Finally, we show how
the transformations between classical reference frames described by the
standard Galilei group symmetries can be obtained from the group of
transformations between quantum reference frames by taking the zero limit of
the parameter that governs the additional noncommutativity introduced by the
quantum nature of inertial transformations.
