Abstract and Applied Analysis
Volume 2011 (2011), Article ID 919538, 35 pages
doi:10.1155/2011/919538
Research Article

The Lie Group in Infinite Dimension

Department of Mathematics, Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, 602 00 Brno, Czech Republic

Received 6 December 2010; Accepted 12 January 2011

Academic Editor: Miroslava Růžičková

Copyright © 2011 V. Tryhuk et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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, 𝐶 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.