HALF-LIE GROUPS

HALF-LIE GROUPS In this paper we study the Lie theoretic properties of a class of topological groups which carry a Banach manifold structure but whose multiplication is not smooth. If G and N are Banach–Lie groups and π : G → Aut(N) is a homomorphism defining a continuous action of G on N, then H := N ⋊π G is a Banach manifold with a topological group structure for which the left multiplication maps are smooth, but the right multiplication maps need not be. We show that these groups share surprisingly many properties with Banach–Lie groups: (a) for every regulated function ξ : [0, 1] → 𝔥 the initial value problem γ ⋅ $$ \overset{\cdot }{\gamma } $$ (t) = γ(t)ξ(t), γ(0) = 1 H , has a solution and the corresponding evolution map from curves in 𝔥 to curves in H is continuous; (b) every C 1-curve γ with γ(0) = 1 and γ′(0) = x satisfies lim n→∞ γ(t/n) n = exp(tx); (c) the Trotter formula holds for C 1 one-parameter groups in H; (d) the subgroup N ∞ of elements with smooth G-orbit maps in N carries a natural Fréchet–Lie group structure for which the G-action is smooth; (e) the resulting Fréchet–Lie group H ∞ := N ∞ ⋊ G is also regular in the sense of (a). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Transformation Groups Springer Journals

HALF-LIE GROUPS

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Publisher
Springer US
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Topological Groups, Lie Groups; Algebra
ISSN
1083-4362
eISSN
1531-586X
D.O.I.
10.1007/s00031-018-9485-6
Publisher site
See Article on Publisher Site

Abstract

In this paper we study the Lie theoretic properties of a class of topological groups which carry a Banach manifold structure but whose multiplication is not smooth. If G and N are Banach–Lie groups and π : G → Aut(N) is a homomorphism defining a continuous action of G on N, then H := N ⋊π G is a Banach manifold with a topological group structure for which the left multiplication maps are smooth, but the right multiplication maps need not be. We show that these groups share surprisingly many properties with Banach–Lie groups: (a) for every regulated function ξ : [0, 1] → 𝔥 the initial value problem γ ⋅ $$ \overset{\cdot }{\gamma } $$ (t) = γ(t)ξ(t), γ(0) = 1 H , has a solution and the corresponding evolution map from curves in 𝔥 to curves in H is continuous; (b) every C 1-curve γ with γ(0) = 1 and γ′(0) = x satisfies lim n→∞ γ(t/n) n = exp(tx); (c) the Trotter formula holds for C 1 one-parameter groups in H; (d) the subgroup N ∞ of elements with smooth G-orbit maps in N carries a natural Fréchet–Lie group structure for which the G-action is smooth; (e) the resulting Fréchet–Lie group H ∞ := N ∞ ⋊ G is also regular in the sense of (a).

Journal

Transformation GroupsSpringer Journals

Published: May 29, 2018

References

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