Infinite-Dimensional Generalizations of Orthogonal Groups over Hilbert Spaces : Constructions and Properties

TL;DR

This paper extends the concept of orthogonal groups to infinite-dimensional Hilbert spaces through two constructions, analyzing their algebraic and topological relationships, and establishing fiber bundle structures involving these groups.

Contribution

It introduces two new infinite-dimensional orthogonal group generalizations and explores their algebraic and topological properties, including subgroup relations and fiber bundle structures.

Findings

$ ext{Theta}( ext{kappa})$ is a topological normal subgroup of $O( ext{kappa})$

The sequence $O^{(n)}( ext{kappa}) o O^{(n+1)}( ext{kappa}) o S^{ ext{kappa}}$ forms a fiber bundle

The paper clarifies the relationship between finite multiplications of reflections and automorphism groups in Hilbert spaces.

Abstract

In real Hilbert spaces, this paper generalizes the orthogonal groups $\mathrm{O}(n)$ in two ways. One way is by finite multiplications of a family of operators from reflections which results in a group denoted as $\Theta(\kappa)$, the other is by considering the automorphism group of the Hilbert space denoted as $O(\kappa)$. We also try to research the algebraic relationship between the two generalizations and their relationship to the stable~orthogonal~group~$\mathrm{O}=\varinjlim\mathrm{O}(n)$ in terms of topology. In this paper we mainly show that : (a) $\Theta(\kappa)$ is a topological and normal subgroup of $O(\kappa)$; (b) $O^{(n)}(\kappa) \to O^{(n+1)}(\kappa) \stackrel{\pi}{\to} S^{\kappa}$ is a fibre bundle where $O^{(n)}(\kappa)$ is a subgroup of $O(\kappa)$ and $S^{\kappa}$ is a generalized sphere.