Lie algebras of quotient groups

TL;DR

This paper establishes conditions under which quotient groups of diffeological groups form Lie algebras, extending the Lie functor to infinite-dimensional and elastic diffeological spaces, and applies this to integrate certain Banach-Lie algebras.

Contribution

It introduces a framework for Lie algebras of quotient groups in diffeological spaces, including infinite-dimensional cases, and characterizes convenience via the diffeological tangent functor.

Findings

Conditions for Lie algebra structures on quotient groups are provided.

Convenient infinite-dimensional manifolds are characterized as elastic diffeological spaces.

Some Banach-Lie algebras are integrated into diffeological groups.

Abstract

We give conditions on a diffeological group $G$ and a normal subgroup $H$ under which the quotient group $G/H$ differentiates to a Lie algebra for which $\operatorname{Lie}(G/H) \cong \operatorname{Lie}(G)/\operatorname{Lie}(H)$. Our Lie functor is instantiated by the tangent structure on elastic diffeological spaces introduced by Blohmann. The requisite conditions on $G$ and $H$ hold, for example, when $G$ is a convenient infinite-dimensional Lie group and $H$ is countable, or when $G$ is finite-dimensional and $H$ is arbitrary. To recognize that convenient infinite-dimensional manifolds are elastic diffeological spaces, we give a characterization of convenience in terms of the diffeological tangent functor: a separated and bornological locally convex topological vector space $E$ is convenient if and only if the natural map $E \times E \to TE$ is an isomorphism of diffeological spaces. As an application, we integrate some classically non-integrable Banach-Lie algebras to diffeological groups.