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Mathematics - Algebra | Infinite-dimensional Lie Groups. General Theory and Main Examples

Infinite-dimensional Lie Groups. General Theory and Main Examples

Series: Graduate Texts in Mathematics, Vol. 274

Glöckner, Helge, Neeb, Karl-Hermann

2016, 350 p. 10 illus.

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  • About this textbook

  • Provides a comprehensive introduction to this important subject, examining the basic structure theory of infinite-dimensional Lie groups
  • Essentially self-contained, provides all necessary background, excepting modest prerequisites
  • Clear exposition includes careful explanations, illustrative examples, numerous exercises, and detailed cross-references to simplify a non-linear reading of the material

Symmetries play a decisive role in the natural sciences and throughout mathematics. Infinite-dimensional Lie theory deals with symmetries depending on infinitely many parameters. Infinite-dimensional Lie Groups provides a comprehensive introduction to this important subject by developing a global infinite-dimensional Lie theory on the basis that a Lie group is simply a manifold modeled on a locally convex space, equipped with a group structure with smooth group operations. The focus is on the local and global level, as well as on the translation mechanisms allowing or preventing passage between Lie groups and Lie algebras. Starting from scratch, the reader is led from the basics of the theory through to the frontiers of current research.

This introductory volume subtitled, General Theory and Main Examples, examines the structure theory of infinite-dimensional Lie groups by developing a broad framework of Lie theory and illustrating the general results through a detailed discussion of the major classes of Lie groups: linear Lie groups, groups of (smooth) maps, groups of diffeomorphisms, and direct limit groups. From these, most other relevant groups can be obtained as extensions or Lie subgroups.

Together with its companion volume subtitled, Geometry and Topology, these essentially self-contained texts provide all necessary background as regards generally locally convex spaces, finite-dimensional Lie theory and differential geometry, with modest prerequisites limited to a basic knowledge of abstract algebra, point set topology, differentiable manifolds, and functional analysis in Banach spaces. The clear exposition includes careful explanations, illustrative examples, numerous exercises, and detailed cross-references to simplify a non-linear reading of the material.

Content Level » Research

Related subjects » Algebra - Analysis - Geometry & Topology - Theoretical, Mathematical & Computational Physics

Table of contents 

Preface.- Introduction.- Infinite-dimensional Calculus.- Infinite-dimensional Manifolds.- Lie Groups.- Locally Exponential Lie Groups.- Linear Lie Groups.- Direct Limits of Lie Groups.- Groups of Maps.- Groups of Diffeomorphisms.- Appendix A: Tools from Topology.- Appendix B: Basic Theory of Locally Convex Spaces.- Appendix C: Finite-dimensional Lie Algebras.- Appendix D: Calculus in Banach Spaces.- Appendix E: Smooth Maps into non-Lie Groups.- Appendix F: Cohomology of Lie Algebras.- Bibliography.- Index.

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