Developments and Trends in Infinite-Dimensional Lie Theory

✍ Scribed by Bruce Allison, John Faulkner (auth.), Karl-Hermann Neeb, Arturo Pianzola (eds.)

• Publisher: Birkhäuser Basel
• Year: 2011
• Tongue: English
• Leaves: 491
• Series: Progress in Mathematics 288
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-viii
Front Matter....Pages 1-1
Isotopy for Extended Affine Lie Algebras and Lie Tori....Pages 3-43
Remarks on the Isotriviality of Multiloop Algebras....Pages 45-51
Extended Affine Lie Algebras and Other Generalizations of Affine Lie Algebras – A Survey....Pages 53-126
Tensor Representations of Classical Locally Finite Lie Algebras....Pages 127-150
Lie Algebras, Vertex Algebras, and Automorphic Forms....Pages 151-168
Kac–Moody Superalgebras and Integrability....Pages 169-218
Front Matter....Pages 219-219
Jordan Structures and Non-Associative Geometry....Pages 221-241
Direct Limits of Infinite-Dimensional Lie Groups....Pages 243-280
Lie Groups of Bundle Automorphisms and Their Extensions....Pages 281-338
Gerbes and Lie Groups....Pages 339-364
Front Matter....Pages 365-365
Functional Analytic Background for a Theory of Infinite-Dimensional Reductive Lie Groups....Pages 367-392
Heat Kernel Measures and Critical Limits....Pages 393-415
Coadjoint Orbits and the Beginnings of a Geometric Representation Theory....Pages 417-457
Infinite-Dimensional Multiplicity-Free Spaces I: Limits of Compact Commutative Spaces....Pages 459-481
Back Matter....Pages 483-492

✦ Subjects

Topological Groups, Lie Groups; Group Theory and Generalizations; Algebra; Geometry; Algebraic Geometry