Lie groups over non-discrete topological fields

TL;DR

This paper extends the theory of infinite-dimensional Lie groups to non-discrete topological fields, developing new differential calculus tools and analyzing various classes of Lie groups in this broader setting.

Contribution

It introduces a generalized framework for Lie groups over non-discrete topological fields, including new differentiability results and techniques applicable to non-locally convex spaces.

Findings

Established differentiability properties of composition and evaluation

Developed exponential laws for function spaces

Provided techniques for non-linear mappings in test function spaces

Abstract

We generalize the classical construction principles of infinite-dimensional real (and complex) Lie groups to the case of Lie groups over non-discrete topological fields. In particular, we discuss linear Lie groups, mapping groups, test function groups, diffeomorphism groups, and weak direct products of Lie groups. The specific tools of differential calculus required for the Lie group constructions are developed. Notably, we establish differentiability properties of composition and evaluation, as well as exponential laws for function spaces. We also present techniques to deal with the subtle differentiability and continuity properties of non-linear mappings between spaces of test functions. Most of the results are independent of any specific properties of the topological vector spaces involved; in particular, we can deal with real and complex Lie groups modeled on non-locally convex spaces.