2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/173319Contents * 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 $(\Gtm,Θ)$ * 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 Schrödinger group and Quantum Optics -- The Virasoro group and String Theory52 pages, latex, no figures. Contribution to "Symmetries in Science X", held in Bregenz (Austria), 13-18 July 1997. To appear in the proceedingsMathematical PhysicsHigh Energy Physics - Theorytext