On smooth infinite dimensional grassmannians, splittings and non-commutative generalized cross-ratio mappings

TL;DR

This paper explores the diffeological structures of infinite-dimensional Grassmannians, extends the concept of non-commutative cross-ratio, and demonstrates their smoothness with various examples, advancing the mathematical understanding of infinite-dimensional geometry.

Contribution

It introduces a rigorous diffeological framework for infinite-dimensional Grassmannians and non-commutative cross-ratios, providing new tools for their analysis.

Findings

Established smoothness of non-commutative cross-ratio

Provided examples from Banach Grassmannians

Developed foundational diffeological structures for infinite-dimensional spaces

Abstract

We describe basic diffeological structures related to splittings and Grassmannians for infinite dimensional vector spaces. We analyze and expand the notion of non-commutative cross-ratio and prove its smoothness. Then we illustrate this theory by examples, with some of them extracted from the existing literature related to infinite dimensional (Banach) Grassmannians, and others where the diffeological setting is a key primary step for rigorous definitions.