Quantum groups and their applications in nuclear physics

Quantum algebras are a mathematical tool which provides us with a class of symmetries wider than that of Lie algebras, which are contained in the form er as a special case. After a self-contained introduction to the necessary mathematical tools (q-numbers, q-analysis, q-oscillators, q-algebras), the su(q)(2) rotator model and its extensions, the construction of deformed exa ctly soluble models (u(3)superset of so(3) model, Interacting Boson Model, Moszkowski model), the 3-dimensional q-deformed harmonic oscillator and its relation to the nuclear shell model, the use of deformed bosons in the des cription of pairing correlations, and the symmetries of the anisotropic qua ntum harmonic oscillator with rational ratios of frequencies, which underly the structure of superdeformed and hyperdeformed nuclei, are discussed in some detail. A brief description of similar applications to the structure o f molecules and of atomic clusters, as well as an outlook are also given.