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# Topics in the History of Lie Groups and Algebraic Groups - Volume 21

184 pages, parution le 17/12/2001

## Résumé

Lie groups and algebraic groups are important in many major areas of mathematics and mathematical physics. We find them in diverse roles, notably as groups of automorphisms of geometric structures, as symmetries of differential systems, or as basic tools in the theory of automorphic forms. The author looks at their development, highlighting the evolution from the almost purely local theory at the start to the global theory that we know today. Starting from Lie's theory of local analytic transformation groups and early work on Lie algebras, he follows the process of globalization in its two main frameworks: differential geometry and topology on one hand, algebraic geometry on the other. Chapters II to IV are devoted to the former, Chapters V to VIII, to the latter.

The essays in the first part of the book survey various proofs of the full reducibility of linear representations, the contributions of H. Weyl to representations and invariant theory for semisimple Lie groups, and conclude with a chapter on E. Cartan's theory of symmetric spaces and Lie groups in the large.

The second part of the book first outlines various contributions to linear algebraic groups in the 19th century, due mainly to E. Study, E. Picard, and above all, L. Maurer. After being abandoned for nearly fifty years, the theory was revived by C. Chevalley and E. Kolchin, and then further developed by many others. This is the focus of Chapter VI. The book concludes with two chapters on the work of Chevalley on Lie groups and Lie algebras and of Kolchin on algebraic groups and the Galois theory of differential fields, which put their contributions to algebraic groups in a broader context.

Professor Borel brings a unique perspective to this study. As an important developer of some of the modern elements of both the differential geometric and the algebraic geometric sides of the theory, he has a particularly deep understanding of the underlying mathematics. His lifelong involvement and his historical research in the subject area give him a special appreciation of the story of its development.

Copublished with the London Mathematical Society. Members of the LMS may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners.

Contents

• Overview
• Full reducibility and invariants for $\mathbf{SL}_2(\mathbb C)$
• Hermann Weyl and Lie groups
• Élie Cartan, symmetric spaces and Lie groups
• Linear algebraic groups in the 19th century
• Linear algebraic groups in the 20th century
• The work of Chevalley in Lie groups and algebraic groups
• Algebraic groups and Galois theory in the work of Ellis R. Kolchin
• Name index
• Subject index
• Photo section

## Caractéristiques techniques

 PAPIER Éditeur(s) American Mathematical Society (AMS) Auteur(s) Armand Borel Parution 17/12/2001 Nb. de pages 184 Format 18 x 26 Couverture Relié Poids 550g Intérieur Noir et Blanc EAN13 9780821802885 ISBN13 978-0-8218-0288-5

## Nouveautés "Algèbre et groupes de lie"

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