Is there a $k \in \mathbb{N}\cup \{\infty\}$ such that $C^k(\mathbb{R})$ forms an infinite-dimensional Lie group? In general I'm assuming they form a Lie-Groupoid with point-wise multiplication?
If so, for which of these $k$ is there a well-developed theory and where could I find some references?
Thank you.
Infinite-dimensional Lie groups are usually required to be Banach manifolds, which $C^k(\mathbb{R})$ isn't. In any case, pointwise multiplication cannot be used to give a group structure, unless you only take the non-vanishing functions (and get an abelian topological group). Pointwise addition gives $C^k(\mathbb{R})$ the structure of an abelian topological group.
Also, pointwise multiplication does not make $C^k(\mathbb{R})$ into a groupoid, as there is no object inclusion map.