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 onℝ∞) 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.
Highlights
A Lie group acting on finite-dimensional space is generated by its infinitesimal transformations and any Lie algebra of vector fields in finite dimension generates a Lie group the first fundamental theorem
We prove that the local, C∞ smooth action of a Lie group on infinite-dimensional space a manifold modelled on Ê∞ 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
Summary
A Lie group acting on finite-dimensional space is generated by its infinitesimal transformations and 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. The result is applied to the theory of generalized or higher-order infinitesimal symmetries of differential equations
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