Group Contractions via Infinite-Dimensional Lie Theory
Contractions are a procedure to construct a new Lie algebra out of a given one via a singular limit. Specifically, the \.{I}n\"on\"u--Wigner construction starts with a Lie algebra $\mathfrak{g}$ with Lie subalgebra $\mathfrak{h} \subseteq \mathfrak{g}$ and complement $\mathfrak{n}$. Then, the vectors in $\mathfrak{n}$ are rescaled by a formal parameter $\varepsilon \in \mathbb{R}_+$, which effectively turns the Lie bracket $[ \, \cdot \, , \cdot \, ]$ into an $\varepsilon$-dependent family $[ \, \cdot \, , \cdot \, ]_\varepsilon$. Notably, the limit $\varepsilon \to 0$ trivializes certain relations, such that the complement $\mathfrak{n}$ becomes an abelian ideal. In the present article, we are not only interested in the limiting Lie algebras and groups, but also in the corresponding power series expansions in $\varepsilon$ to understand their limiting behavior. Particularly, we are interested in the integration of the `power-series-expanded' Lie algebras to their corresponding Lie groups. To this end, we reformulate the above procedure using infinite-dimensional Lie algebras of analytic germs and then apply their integration theory. Our main results are a construction of the corresponding `Lie group expansions' in terms of quotients of groups of analytic germs and an explicit description of these groups in elementary terms. Applications of this procedure include the geometric Newtonian limit of General Relativity to Newton--Cartan gravity, where the Poincar\'e group is contracted to the Galilei group.