The Lie Group in Infinite Dimension

Abstract EN

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 ℝ∞) 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.