Introduction

Summary

La Nature est un temple où de vivants piliers

Laissent parfois sortir de confuses paroles;

L'homme y passe à travers des forêts de symboles

Qui l'observent avec des regards familiers.

Charles Baudelaire, Les Fleurs du Mal

И я выхожу из пространства

В запущеннь сад величин

Osip Mandelstam

Introduction

§0.1. The creators of the Lie theory viewed a Lie group as a group of symmetries of an algebraic or a geometric object; the corresponding Lie algebra, from their point of view, was the set of infinitesimal transformations. Since the group of symmetries of the object is not necessarily finite-dimensional, S. Lie considered not only the problem of classification of subgroups of GL_n, but also the problem of classification of infinite-dimensional groups of transformations.

The problem of classification of simple finite-dimensional Lie algebras over the field of complex numbers was solved by the end of the 19th century by W. Killing and E. Cartan. (A vivid description of the history of this discovery, one of the most remarkable in all of mathematics, can be found in Hawkins [1982].) And just over a decade later, Cartan classified simple infinite-dimensional Lie algebras of vector fields on a finite-dimensional space.

Starting with the works of Lie, Killing, and Cartan, the theory of finite-dimensional Lie groups and Lie algebras has developed systematically in depth and scope. On the other hand, Cartan's works on simple infinite-dimensional Lie algebras had been virtually forgotten until the mid-sixties. A resurgence of interest in this area began with the work of Guillemin–Stemberg [1964] and Singer–Sternberg [1965], which developed an adequate algebraic language and the machinery of filtered and graded Lie algebras.