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Direct limits of regular Lie groups
Author(s) -
Glöckner Helge
Publication year - 2021
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201900073
Subject(s) - mathematics , lie group , simple lie group , representation of a lie group , pure mathematics , paracompact space , maximal torus , adjoint representation , direct limit , limit (mathematics) , group of lie type , fundamental representation , mathematical analysis , lie algebra , group theory , hausdorff space , weight
Let G be a regular Lie group which is a directed union of regular Lie groups G i (all modelled on possibly infinite‐dimensional, locally convex spaces). We show that G = lim ⟶G ias a regular Lie group if G admits a so‐called direct limit chart. Notably, this allows the regular Lie groupDiff c ( M )of compactly supported diffeomorphisms to be interpreted as a direct limit of the regular Lie groupsDiff K ( M )of diffeomorphisms supported in compact sets K ⊆ M , even if the finite‐dimensional smooth manifold M is merely paracompact (but not necessarily σ‐compact), which is new. Similar results are obtained for the test function groupsC c k ( M , F )with values in a Lie group F .