The Lie Group in Infinite Dimension

V. Tryhuk, V. Chrastinová and O. Dlouhý

Abstract and Applied Analysis, 2011, vol. 2011, 1-35

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, ð ¶ âˆž 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.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:919538

DOI: 10.1155/2011/919538

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