Symmetries of the Quantum State Space and Group Representations

The homomorphisms of a connected Lie group G into the symmetry group of a quantum system are classified in terms of unitary representations of a simply connected Lie group associated with G. Moreover, an explicit description of the T-multipliers of G is obtained in terms of the R-multipliers of the universal covering G* of G and the characters of G*. As an application, the Poincare group and the Galilei group, both in 3 + 1 and 2 + 1 dimensions, are considered.