Group Contractions via Infinite-Dimensional Lie Theory

Abstract: Contractions are a procedure to construct a new Lie algebra out of a given one via a singular limit. Specifically, the İnönü--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 $\mathfrak{h}$ are rescaled by a formal parameter $\varepsilon \in \mathbb{R}_+$, which effectively turns the Lie bracket $[ \, \cdot \, , \cdot \, ]$ into a formal power series. Notably, the limit $\varepsilon \to 0$ trivialises 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 series expansions in $\varepsilon$ to understand the limiting behaviour. Particularly, we are interested in how to integrate the `power-series-expanded' Lie algebras to the Lie group level. To this end, we reformulate the above procedure using infinite-dimensional Lie algebras of analytic germs. Then, we apply their integration theory to obtain an extensive analysis of this expansion procedure. In particular, we obtain an explicit construction of the resulting Lie algebras and groups.