2025/12/20 by David Prinz, Prinz, David, Alexander Schmeding +3
Mathematics · #17B65 #17B81 #22E65 (Primary) #22E70 #58B25 (Secondary) #Advanced Algebra and Geometry #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2512.18530
openalex publication_date 2025/12/20 · openalex created_date 2025/12/24 · openalex updated_date 2026/07/28
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 \mathfrakg with Lie subalgebra \mathfrakh ⊆ \mathfrakg and complement \mathfrakn. Then, the vectors in \mathfrakn are rescaled by a formal parameter ε ∈ ℝ+, which effectively turns the Lie bracket [ ⋅ , ⋅ ] into an ε-dependent family [ ⋅ , ⋅ ]ε. Notably, the limit ε → 0 trivializes certain relations, such that the complement \mathfrakn 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 ε 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 Poincaré group is contracted to the Galilei group.