Symmetric bundles and representations of Lie triple systems

TL;DR

This paper introduces symmetric bundles as geometric analogs of Lie triple system representations, exploring their properties and the relationship between differential geometry and representation theory.

Contribution

It defines symmetric bundles in the context of symmetric spaces and investigates their connection to Lie triple system representations and reflection spaces.

Findings

Symmetric bundles correspond to representations of Lie triple systems.

The forgetful functor from symmetric bundles to reflection spaces is not injective.

Unusual symmetric bundle structures can exist on tangent bundles of symmetric spaces.

Abstract

We define symmetric bundles as vector bundles in the category of symmetric spaces; it is shown that this notion is the geometric analog of the one of a representation of a Lie triple system. We show that such a bundle has an underlying reflection space, and we investigate the corresponding forgetful functor both from the point of view of differential geometry and from the point of view of representation theory. This functor is not injective, as is seen by constructing "unusual" symmetric bundle structures on the tangent bundles of certain symmetric spaces.