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Derivative of the exponential mapping for infinite dimensional Lie groups. (English) Zbl 0836.22028

Summary: It is proved that for infinite dimensional Lie groups in the sense of the differential calculus of Frölicher and Kriegl the derivative of the exponential mappings is given by the formula \[ \text{d (exp)} (X) Y = d\lambda_{\text{exp}(X)} (e) \int^1_0 \text{Ad}_{\text{exp} (-tX)} Y, \] where \(\lambda\) stands for the left translation and \(e\) is the neutral element.

MSC:

22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
58B25 Group structures and generalizations on infinite-dimensional manifolds
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