  Contents
  * Introduction
  -- Why $S^1$-extended phase space?
  -- Why central extensions of classical symmetries?
  * Central extension \Gt of a group $G$
  -- Group cohomology
  -- Cohomology and contractions: Pseudo-cohomology
  -- Principal bundle with connection $(\Gtm,\Theta)$
  * Group Approach to Quantization
  -- $U(1)$-quantization
  -- Non-horizontal polarizations
  * Simple examples
  -- The abelian group $R^{k}$
  -- The semisimple group $SU(2)$
  * Algebraic anomalies
  -- Higher-order polarizations
  -- The Schr\"odinger group and Quantum Optics
  -- The Virasoro group and String Theory
