Department of Mathematics, Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, 602 00 Brno, Czech Republic
Copyright © 2011 V. Tryhuk et al. This is an open access article distributed under the
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A Lie group acting on finite-dimensional space is generated by its infinitesimal
transformations and conversely, any Lie algebra of vector fields in finite dimension
generates a Lie group (the first fundamental theorem). This classical result is adjusted
for the infinite-dimensional case. We prove that the (local, smooth) action
of a Lie group on infinite-dimensional space (a manifold modelled on ) may be
regarded as a limit of finite-dimensional approximations and the corresponding Lie
algebra of vector fields may be characterized by certain finiteness requirements. The
result is applied to the theory of generalized (or higher-order) infinitesimal symmetries
of differential equations.