Lie algebras of quotient 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, and $\operatorname{Lie}(G/H) \cong \operatorname{Lie}(G)/\operatorname{Lie}(H)$. Our Lie functor is derived from the tangent structure on elastic diffeological spaces introduced by Blohmann. The requisite conditions on $G$ and $H$ hold when $G$ is a convenient infinite-dimensional Lie group and all iterated tangent bundles $T^kH$ are initial subgroups of $T^kG$; in particular, $G$ may be finite-dimensional, or $H$ may be countable. As an application, we integrate some classically non-integrable Banach-Lie algebras to diffeological groups.