Infinite Dimensional Lie Algebras An Introduction

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Infinite Dimensional Lie Algebras: An Introduction by Victor G. Kac
English | PDF | 1983 | 267 Pages | ISBN : 1475713843 | 18.3 MB
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.
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