Group Contractions via Infinite-Dimensional Lie Theory

TL;DR

This paper explores the process of Lie algebra contractions via infinite-dimensional Lie theory, providing a detailed analysis of series expansions and their integration to Lie groups, with explicit constructions and new insights.

Contribution

It reformulates Lie algebra contractions using infinite-dimensional analytic germs and develops an explicit method to integrate these series expansions into Lie groups.

Findings

Explicit construction of contracted Lie algebras and groups

Analysis of series expansion behavior in contractions

Application of infinite-dimensional Lie theory to contraction procedures

Abstract

Contractions are a procedure to construct a new Lie algebra out of a given one via a singular limit. Specifically, the \.In\"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 $\mathfrak{h}$ 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.