Group Contractions via Infinite-Dimensional Lie Theory
David Prinz, Alexander Schmeding, Philip K. Schwartz
TL;DR
The paper addresses how to integrate power-series-expanded Lie algebras, arising from İnönü–Wigner contractions, at the Lie group level. It recasts contractions as analytic germ contractions and defines Lie algebra expansions $\mathfrak g_\phi^{(k)}$ that encode higher-order approach to the limit, then lifts this to the group level by working with germ groups $\Gamma(G)$ and the subgroup $\Gamma(G)_H$, proving that the $k$-th group expansion $G^{(k)}_H$ is a BCH Banach Lie group with Lie algebra $\mathfrak g^{(k)}_\mathfrak h$ and is analytically isomorphic to a semidirect product $H \ltimes_\mathrm{Ad} (\mathfrak g^k \times (\mathfrak g/\mathfrak h), \circledast)$. The main results include an explicit lifting theorem and a concrete first-order expansion formula, demonstrated in the classic $\mathrm{SO}(3) \to \mathrm{ISO}(2)$ contraction. This framework provides a rigorous analytic tool for studying symmetry contractions and their higher-order deformations, with potential extensions to graded/ super structures, $L_\infty$-algebras, and algebroid/groupoid settings, offering new avenues for mathematical physics and symmetry analysis.
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.