You are here

The Geometry of Infinite-Dimensional Groups

Boris A. Khesin and Robert Wendt
Publication Date: 
Number of Pages: 
Ergebnisse der Mathematik und ihrer Grenzgebiete 51
[Reviewed by
Fabio Mainardi
, on

Lie groups are one of the most important structures in mathematics. They lie at the foundation of many branches of pure and applied mathematics: number theory, differential equations, ergodic theory, knot theory, string theory, etc. Unlike the so-called classical groups (general linear groups, symplectic, orthogonal and so on), however, a lot of “natural” Lie groups are infinite-dimensional. Consider for instance the group of diffeomorphisms of a compact finite-dimensional manifold, or the group of gauge transformations of a principal bundle over a manifold, or the group of  pseudo differential operators.

In the infinite-dimensional case, the notion of differentiability is defined in terms of Fréchet spaces, i.e., complete locally convex Hausdorff metric spaces, or also Banach spaces. While the basic definitions can be carried through (tangent space, derivations, Lie algebra), some of the fundamental theorems on finite-dimensional Lie groups fail to generalize. For example, the exponential map doesn’t need to exist, and when it does, it may not be a local diffeomorphism. And an infinite-dimensional Lie algebra is not necessarily attached to a Lie group. Such pathologies make impossible, at present, to have a unified vision of infinite-dimensional Lie groups; instead, one has to study separately certain classes and this book is an overview of the most popular and remarkable: diffeomorphisms of the circle, groups of volume-preserving diffeomorphisms, loop groups, pseudo-differential operators.

This book is therefore a kind of mathematical bestiarium, a compendium of fabulous mathematical beasts. Needless to say, it is not an easy book; it is addressed to those mathematicians familiar with the finite-dimensional Lie theory, and with a strong background on differential geometry.

Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are mainly Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. At present, he works in a "classe préparatoire" in Geneva. He may be reached at