Completeness of Infinite-dimensional Lie Groups in Their Left Uniformity

Abstract

Abstract. We prove completeness for the main examples of in nite-dimensional Lie groups and some related topological groups. Consider a sequence G ⊆ G ⊆ ⋅ ⋅ ⋅ of topological groups G n such that G n is a subgroup of G n+ and the latter induces the given topology on G n , for each n ∈ N. Let G be the direct limit of the sequence in the category of topological groups. We show that G induces the given topology on each G n whenever ⋃ n∈N V V ⋅ ⋅ ⋅ V n is an identity neighbourhood in G for all identity neighbourhoods V n ⊆ G n . If, moreover, each G n is complete, then G is complete. We also show that the weak direct product ⊕ j∈J G j is complete for each family (G j ) j∈J of complete Lie groups G j . As a consequence, every strict direct limit G = ⋃ n∈N G n of nite-dimensional Lie groups is complete, as well as the di eomorphism group Di c (M) of a paracompact nite-dimensional smooth manifold M and the test function group C k c (M, H), for each k ∈ N ∪ {∞} and complete Lie group H modelled on a complete locally convex space.

Direct limits of regular Lie groups

“…If M is a paracompact finite‐dimensional smooth manifold, F is a Lie group with neutral element e and , then the “test tunction group” is a Lie group, comprising all ‐maps such that for off some compact set [5, 9, 11, 12]. For fixed K , is a Lie subgroup of .…”

“…Among other things, weak direct products are useful tools for the study of diffeomorphism groups and test functions groups (cf. [7, 9, 12], and [13]). Theorem (Direct limit properties of prime examples).…”