Quantum Groups and Q-lattices in Phase Space

Quantum groups lead to an algebraic structure that can be realized on quantum spaces. These are noncommutative spaces that inherit a well deened mathematical structure from the quantum group symmetry. In turn such quantum spaces can be interpreted as noncom-mutative connguration spaces for physical systems which carry a symmetry like structure. These connguration spaces will be generalized to noncommutative phase space. The definition of the noncommutative phase space will be based on a diierential calculus on the connguration space which is compatible with the symmetry. In addition a conjugation operation will be deened which will allow us to deene the phase space variables in terms of algebraically selfadjoint operators. An interesting property of the phase space observables will be that they will have a discrete spectrum. These noncommutative phase space puts physics on a lattice structure.