Noncommutative Spaces and Quantum Groups

Quantum mechanics has established a noncommutative structure between coordinates and momenta. We might try to go one step further and introduce a noncommutative structure among the coordinates as well. To study the consequences of such an assumption we have to study models. These should be models that allow a very explicit and detailed treatment. From experience we know that this will be the case if we impose additional algebraic structures such as symmetries. It turns out that quantum groups lend themselves very favourably to noncommutative structures of their comodule spaces. Thus it seems very natural to start from a quantum group — a q-deformation of an established symmetry group in physics such as the rotation group or the Lorentz group — study its comodules and the algebraic structures compatible with the quantum groups. These we shall call quantum spaces. We shall focus our interest on such spaces which allow a conjugation compatible with the algebraic structure, as we are interested in something like “real spaces”.

The relations for the elements of the quantum space will be assumed to be homogeneous and quadratic. Next we shall try to mimic a Heisenberg algebra, these are relations between noncommutative coordinates and noncommutative momenta, allowing “real” momenta compatible with the quantum group as well. This then will be the setting of a kinematics of a q-deformed quantum mechanics.

The simplest model which is at the basis of all the modules studied up to now will now be discussed in this lecture. We restrict ourselves to the algebraic properties.