A model of the universal covering group of SO(H,2)

by R. J. Plymen and R. F. Streater; Bull. London Math. Soc., 7, 283-288, 1974.

ABSTRACT

Let H be a separable real Hilbert space. If dim H is finite, then SO(H), the group of orthogonal maps from H onto itself with determinant 1, is a Lie group. The group SO(H) is doubly connected, and its universal covering group, Spin(H), has a realisation as a group in CL(H), the real Clifford algebra over H. In this paper, we obtain a related result when dim H=infty. We explicitly construct the connected,simply connected, universal covering group Spin(H,2) of the Hilbert-Lie group SO(H,2). Here, the 2 refers to the Hilbert-Schmidt norm. Our result confirms a conjecture of P. de la Harpe. The well-known result of Stinespring on this group does not include a proof that the covering map is a local homeomorphism. In particular, Stinespring did not prove that the inverse, defined locally, is continuous. This illustrates one difficulty of analysis in infinite dimensions: the inverse of a continuous bijection might not be continuous.

P. de la Harpe has obtained a similar result for the smaller Banach-Lie group SO(H,1), where now 1 refers to the trace-norm.

Recently, more results on these groups have been obtained by Maria Gordina.

A nice account of the history of Clifford algebras can be found here, where our paper is almost mentioned.