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This book is intended for graduate students and researchers interested in the general mathematical framework of projective representations and its application to groups that are of interest in the study of physical quantum symmetries. Projective representations are required for quantum symmetries as physical transition probabilities between physical states in quantum theory are given by the square of the modulus of the states. This results in a phase degree of freedom, the quantum phase, that manifests in the symmetries through projective representations. 
Part I of the book is a primer of the mathematical theory required for the study of symmetries.  Lie groups (and in particular, matrix groups) are reviewed and the general properties of their representations are studied.   Finally, the theory of projective representations is developed for connected Lie groups. 
Part II studies the projective representations to the inhomogeneous Lorentz group that describes the inertial states of special relativistic quantum mechanics and then studies the projective representations of the Galilei group for the ‘nonrelativistic’ limit.   
Part III studies the Weyl-Heisenberg group and its origin in the projective representations of the inhomogeneous symplectic group that is a basic symmetry of the Hamilton formulation of mechanics. The Weyl-Heisenberg group appears as the nonabelian normal subgroup of the central extension of the inhomogeneous symplectic group that is required for the projective representations.   This shows how the Heisenberg commutation relations are a consequence of the transition probabilities being given by the square of the modulus of states and the resulting quantum phase.