A Lie algebra for Fr\"olicher groups

TL;DR

This paper develops a Lie algebra structure for the tangent space at the identity of Fr"olicher groups, which are a broad class of smooth-like groups within a cartesian closed category, extending classical Lie theory.

Contribution

It introduces a Lie algebra framework for Fr"olicher groups, generalizing the tangent space and Lie bracket concepts beyond traditional smooth manifolds.

Findings

Tangent space at the identity can be equipped with a Lie bracket under certain conditions

Fr"olicher spaces form a cartesian closed category including many infinite-dimensional groups

An example satisfying the additional assumption is discussed

Abstract

Fr\"olicher spaces form a cartesian closed category which contains the category of smooth manifolds as a full subcategory. Therefore, mapping groups such as C^\infty(M,G) or \Diff(M), but also projective limits of Lie groups are in a natural way objects of that category, and group operations are morphisms in the category. We call groups with this property Fr\"olicher groups. One can define tangent spaces to Fr\"olicher spaces, and in the present article we prove that, under a certain additional assumption, the tangent space at the identity of a Fr\"olicher group can be equipped with a Lie bracket. We discuss an example which satisfies the additional assumption.