  Quantum symmetries that leave invariant physical transition probabilities are
described by projective representations of Lie groups. The mathematical theory
of projected representations for topologically connected Lie groups is reviewed
and applied to the inhomogeneous symplectic group ISp(2n). The projective
representations are given in terms of the ordinary unitary representations of
the central extension of the group with respect to its first homotopy group
direct product with its second cohomology group. The second cohomology group of
ISp(2n) is the one dimensional abelian group and the central extension turns
the 2n dimension abelian normal subgroup of translations of ISp(2n) into the
Weyl-Heisenberg group. This is the quantum symmetry origin of the Heisenberg
commutation relations that are the Hermitian representation of the Lie algebra
of the Weyl-Heisenberg group.
