Publication result detail

The Lie Group in Infinite Dimension

TRYHUK, V.; CHRASTINOVÁ, V.; DLOUHÝ, O.

Original Title

The Lie Group in Infinite Dimension

English Title

The Lie Group in Infinite Dimension

Type

Peer-reviewed article not indexed in WoS or Scopus

Original 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^\infty smooth) action of a Lie group on infinite-dimensional space (a manifold modelled on R^\infty) 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.

English 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^\infty smooth) action of a Lie group on infinite-dimensional space (a manifold modelled on R^\infty) 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.

Keywords

Lie first main theorem; local one--parameter group; local Lie group; generalized infinitesimal symmetries; diffiety

Key words in English

Lie first main theorem; local one--parameter group; local Lie group; generalized infinitesimal symmetries; diffiety

Authors

TRYHUK, V.; CHRASTINOVÁ, V.; DLOUHÝ, O.

RIV year

2011

Released

24.02.2011

Publisher

Hindawi Publishing Corporation

Location

USA

ISBN

1085-3375

Periodical

Abstract and Applied Analysis

Volume

2011

Number

1

State

United States of America

Pages from

1

Pages to

35

Pages count

35

Full text in the Digital Library

http://hdl.handle.net/

BibTex

@article{BUT50550,
  author="Václav {Tryhuk} and Veronika {Chrastinová} and Oldřich {Dlouhý}",
  title="The Lie Group in Infinite Dimension",
  journal="Abstract and Applied Analysis",
  year="2011",
  volume="2011",
  number="1",
  pages="1--35",
  issn="1085-3375"
}