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Developments and Trends in Infinite-Dimensional Lie Theory

  • Book
  • © 2011

Overview

Editors:
  1. Karl-Hermann Neeb

    1. Friedrich-Alexander-Universität Erlangen, Department of Mathematics, Erlangen, Germany

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    You can also search for this editor in PubMed   Google Scholar

  2. Arturo Pianzola

    1. University of Alberta, Department of Mathematical Sciences, Edmonton, Canada

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    You can also search for this editor in PubMed   Google Scholar

  • Invited papers written by distinguished researchers
  • Expository essays focus on recent developments and trends in infinite-dimensional Lie theory
  • Discusses new methods, structures, and representations of infinite-dimensional Lie groups
  • Includes supplementary material: sn.pub/extras

Part of the book series: Progress in Mathematics (PM, volume 288)

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-viii
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  2. Infinite-Dimensional Lie (Super-)Algebras

    1. Front Matter

      Pages 1-1
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    2. Isotopy for Extended Affine Lie Algebras and Lie Tori

      • Bruce Allison, John Faulkner
      Pages 3-43

    3. Remarks on the Isotriviality of Multiloop Algebras

      • Philippe Gille, Arturo Pianzola
      Pages 45-51

    4. Extended Affine Lie Algebras and Other Generalizations of Affine Lie Algebras – A Survey

      • Erhard Neher
      Pages 53-126

    5. Tensor Representations of Classical Locally Finite Lie Algebras

      • Ivan Penkov, Konstantin Styrkas
      Pages 127-150

    6. Lie Algebras, Vertex Algebras, and Automorphic Forms

      • Nils R. Scheithauer
      Pages 151-168

    7. Kac–Moody Superalgebras and Integrability

      • Vera Serganova
      Pages 169-218

  3. Geometry of Infinite-Dimensional Lie (Transformation) Groups

    1. Front Matter

      Pages 219-219
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    2. Jordan Structures and Non-Associative Geometry

      • Wolfgang Bertram
      Pages 221-241

    3. Direct Limits of Infinite-Dimensional Lie Groups

      • Helge Glöckner
      Pages 243-280

    4. Lie Groups of Bundle Automorphisms and Their Extensions

      • Karl-Hermann Neeb
      Pages 281-338

    5. Gerbes and Lie Groups

      • Christoph Schweigert, Konrad Waldorf
      Pages 339-364

  4. Representation Theory of Infinite-Dimensional Lie Groups

    1. Front Matter

      Pages 365-365
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    2. Functional Analytic Background for a Theory of Infinite-Dimensional Reductive Lie Groups

      • Daniel Beltiţă
      Pages 367-392

    3. Heat Kernel Measures and Critical Limits

      • Doug Pickrell
      Pages 393-415

    4. Coadjoint Orbits and the Beginnings of a Geometric Representation Theory

      • Tudor S. Ratiu
      Pages 417-457

    5. Infinite-Dimensional Multiplicity-Free Spaces I: Limits of Compact Commutative Spaces

      • Joseph A. Wolf
      Pages 459-481

  5. Back Matter

    Pages 483-492
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Keywords

About this book

This collection of invited expository articles focuses on recent developments and trends in infinite-dimensional Lie theory, which has become one of the core areas of modern mathematics. The book is divided into three parts: infinite-dimensional Lie (super-)algebras, geometry of infinite-dimensional Lie (transformation) groups, and representation theory of infinite-dimensional Lie groups.

Part (A) is mainly concerned with the structure and representation theory of infinite-dimensional Lie algebras and contains articles on the structure of direct-limit Lie algebras, extended affine Lie algebras and loop algebras, as well as representations of loop algebras and Kac–Moody superalgebras.

The articles in Part (B) examine connections between infinite-dimensional Lie theory and geometry. The topics range from infinite-dimensional groups acting on fiber bundles, corresponding characteristic classes and gerbes, to Jordan-theoretic geometries and new results on direct-limit groups.

The analytic representation theory of infinite-dimensional Lie groups is still very much underdeveloped. The articles in Part (C) develop new, promising methods based on heat kernels, multiplicity freeness, Banach–Lie–Poisson spaces, and infinite-dimensional generalizations of reductive Lie groups.

Contributors: B. Allison, D. Beltiţă, W. Bertram, J. Faulkner, Ph. Gille, H. Glöckner, K.-H. Neeb, E. Neher, I. Penkov, A. Pianzola, D. Pickrell, T.S. Ratiu, N.R. Scheithauer, C. Schweigert, V. Serganova, K. Styrkas, K. Waldorf, and J.A. Wolf.

Editors and Affiliations

  • Friedrich-Alexander-Universität Erlangen, Department of Mathematics, Erlangen, Germany

    Karl-Hermann Neeb

  • University of Alberta, Department of Mathematical Sciences, Edmonton, Canada

    Arturo Pianzola

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