The subject of Quantum Groups and Quantum Algebras developed out of ideas in physics in the 80s. Subsequently, the range of applications in physics and their pivotal role in several areas of mathematics has lead to this subject being one of the most active in mathematics. From an algebraic point of view, since many of the algebras involved may be viewed as non commutative deformations of rings of functions on geometric spaces, it has recently become apparent that the subject should be studied as part of the developing theory of Noncommutative Geometry.

In this theory, the noncommutative algebras arising from deformations of the classical commutative case are studied by algebraic means, but from a geometrical perspective. This development is somewhat akin to the development in physics of Quantum Mechanics as a noncommutative deformation of the classical Newtonian view of physics ? the noncommutativity reflecting the uncertainty principle. From this point of view, the "points', "curves', "surfaces', etc. in classical geometry are replaced in the noncommutative geometry by the spectra of prime ideals and primitive ideals and the representation theory of the algebras. The most important algebras that arise in this study are the quantum coordinate algebras and quantum enveloping algebras arising from the classical groups, and, because of its intimate connection with the quantum general and special linear groups, the algebra of quantum matrices.

Important tasks are to understand the spectrum of prime ideals in these and related algebras and to calculate the automorphism groups of the algebras. These are the main tasks involved in this proposal. The two tasks are interlinked. In contrast with their classical counterparts, the quantum deformations are much more rigid objects (at least in the generic case) and this is reflected by the relatively small size of the so-called prime spectrum of these algebras.