© 2011

Developments and Trends in Infinite-Dimensional Lie Theory

  • Karl-Hermann Neeb
  • Arturo Pianzola

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

Table of contents

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

    1. Front Matter
      Pages 1-1
    2. Bruce Allison, John Faulkner
      Pages 3-43
    3. Philippe Gille, Arturo Pianzola
      Pages 45-51
    4. Ivan Penkov, Konstantin Styrkas
      Pages 127-150
    5. Nils R. Scheithauer
      Pages 151-168
    6. Vera Serganova
      Pages 169-218
  3. Geometry of Infinite-Dimensional Lie (Transformation) Groups

    1. Front Matter
      Pages 219-219
    2. Wolfgang Bertram
      Pages 221-241
    3. Helge Glöckner
      Pages 243-280
    4. Christoph Schweigert, Konrad Waldorf
      Pages 339-364
  4. Representation Theory of Infinite-Dimensional Lie Groups

  5. Back Matter
    Pages 483-492

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.


Kac--Moody superalgebras direct limit groups heat kernels loop algebras multiplicity freeness

Editors and affiliations

  • Karl-Hermann Neeb
    • 1
  • Arturo Pianzola
    • 2
  1. 1.Friedrich-Alexander-Universität ErlangenDepartment of MathematicsErlangenGermany
  2. 2.University of AlbertaDepartment of Mathematical SciencesEdmontonCanada

Bibliographic information