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The Lie Group in Infinite Dimension

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  • V. Tryhuk
  • V. Chrastinová
  • O. Dlouhý

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.

Suggested Citation

  • V. Tryhuk & V. Chrastinová & O. Dlouhý, 2011. "The Lie Group in Infinite Dimension," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-35, February.
    • Handle: RePEc:hin:jnlaaa:919538
    DOI: 10.1155/2011/919538
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