We give a classification of the unitary irreducible realizations of the q-oscillator algebra and present their explicit construction. Besides standard representations, we obtain a new one with continuous spectrum absent in the non-deformed case. ∗Part of Project Nr. P8916-PHY of the “Fonds zur Förderung der wissenschaftlichen Forschung in Österreich”.

E(2) is studied as the automorphism group of the Heisenberg algebra H . The basis in the Hilbert space K of functions on H on which the unitary irreducible representations of the group are realized is explicitly constructed. The addition theorem for the Kummer functions is derived. Febriary 2000

A general matrix approach to study entangled states is presented, based on operator completeness relations. Bases of unitary operators are considered, with focus on irreducible representations of groups. Bell measurements for teleportation are considered, and robustness of teleportation to various kinds of non idealities is shown.

Inductive algebras for the irreducible unitary representations of the universal cover of the group of unimodular two by two matrices are classified. The classification of homogeneous shift operators is obtained as a direct consequence. This gives a new approach to the results of Bagchi and Misra.