The “Geometry” Underlying Quantum deSitter Group

Abstract: In the first part of this note, we define a differential geometry on quantum deformed deSitter space. Differential geometry makes sense on a symmetric space because (among other things) the algebra of co-ordinates and their derivatives commutes with the symmetry group of the space. Zumino and others have noticed that if we q-deform the symmetry group, we will need to deform the differential structure on the underlying space as well, in order to preserve an invariant notion of “differential geometry”. We implement this idea to define a quantum version of deSitter space. But to make this definition complete, we need to impose an appropriate reality condition on the quantum group (and thereby on the underlying space). In the second part of this article, we try to constrain the value of the deformation parameter by imposing the condition that in the undeformed limit, we want the real form to be SO(4, 1). Since quantum groups are inherently complex, we need to choose an appropriate ∗-structure (reality condition) to make sure that we are indeed working with SO(4, 1) and not SO(5;C). The choice of ∗-structure that gives rise to SO(4, 1) in the q → 1 limit of the standard q-deformation, exists only when q is a real number. This suggests that trying to construct finite-dimensional deSitter Hilbert spaces by looking at representations of standard deformations with q = root of unity, is not going to work directly. Exotic deformations of the group with non-standard ∗-structures might get beyond this problem.