Algebraic models of many-body systems and their dynamic symmetries and supersymmetries

An overview of the method of spectrum generating algebras (SGA) and dynamical symmetries (DS) is given and applied to a class of models (the so-called s-b boson models) with SGAu (n). Quantum phase transitions (QPT) in these systems are discussed by introducing the coset spaces U(n)/U(n-1)circle times U(1). Finally, spectrum generating superalgebras (SGSA) and dynamical supersymmetries (SUSY) of a class of Bose-Fermi systems with SGAu (n/m) are introduced and applied to s-b Bose-Fermi models.