  This paper explores the use of a deformation by a root of unity as a tool to
build models with a finite number of states for applications to quantum
gravity. The initial motivation for this work was cosmological breaking of
supersymmetry. We explain why the project was unsuccessful. What is left are
some observations on supersymmetry for q-bosons, an analogy between black holes
in de Sitter and properties of quantum groups, and an observation on a
noncommutative quantum mechanics model with two degrees of freedom, depending
on one parameter. When this parameter is positive, the spectrum has a finite
number of states; when it is negative or zero, the spectrum has an infinite
number of states. This exhibits a desirable feature of quantum physics in de
Sitter space, albeit in a very simple, non-gravitational context.
