Quantum states as realizations of groups

A quick review of some Lie algebras related to well-known groups is given. We start with the Heisenberg-Weyl algebra and after the definitions of the Fock states we give the definition of the coherent state of this group. This is followed by the exposition of the SU(2) and SU(1,1) algebras and their coherent states. From there we go on to describe the binomial states and their extensions as the finite dimensional pair coherent states and their nonlinear versions as realizations of the SU(2) group. This is followed by considering the negative binomial states, the single mode and two-mode squeezed states and their variants as realizations of the SU(1,1) group. Generation schemes based on physical systems are considered for some of these states.