**History of Mathematics**

Volume: 21;
2001;
184 pp;
Hardcover

MSC: Primary 01; 17; 20; 22; 32; 53; 57;

Print ISBN: 978-0-8218-0288-5

Product Code: HMATH/21

List Price: $48.00

Individual Member Price: $38.40

**Electronic ISBN: 978-1-4704-3889-0
Product Code: HMATH/21.E**

List Price: $48.00

Individual Member Price: $38.40

# Essays in the History of Lie Groups and Algebraic Groups

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*Armand Borel*

A co-publication of the AMS and the London Mathematical Society

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 of \(\mathbf{SL}_2{(\mathbb{C})}\), 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.

#### Readership

Graduate students and research mathematicians interested in Lie groups and algebraic groups; historians of mathematics.

#### Reviews & Endorsements

The book under review was written by a leading contributor to the development of the theory of Lie and algebraic groups during the second half of the 20th century … Borel's work stands out … by virtue of its genuinely scholarly nature … perceptive essays … Chapter 4 on E. Cartan's theory of symmetric spaces is notable for its detailed analysis of Cartan's main papers on the subject … future historians interested in the interplay of Lie groups and differential geometry will find this essay an invaluable entrée to Cartan's mathematics … Chapter 5 … is especially notable for the new insights it provides into the work of Ludwig Maurer … many historians will benefit greatly from the expertise and insights contained in Borel's thoughtful, carefully written essays.

-- Centaurus, International Magazine of the History of Mathematics, Science and Technology

The exposition in this book is given in a fascinating and understandable language … the book is useful for specialists as well as for people interested in the history of the development of the theory.

-- Mathematica Bohemica

The book is an essential contribution to the conmprhension of the central position taken by Lie group theory ... It should be read by any perosn wanting to learn about the cultural interest of mathematics because it is written by one of the most important actors in the field ...

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Essays in the History of Lie Groups and Algebraic Groups

- Cover Cover11
- Title page v6
- Contents vii8
- Introduction ix10
- Terminology for classical groups and notation xi12
- Photo credits xiii14
- Overview 116
- Full reducibility and invariants for 𝐒𝐋₂(ℂ) 924
- Hermann Weyl and Lie groups 2944
- Élie Cartan, symmetric spaces and Lie groups 5974
- Linear algebraic groups in the 19th century 93108
- Linear algebraic groups in the 20th century 119134
- The work of Chevalley in Lie groups and algebraic groups 147162
- Algebraic groups and Galois theory in the work of Ellis R. Kolchin 149164
- Name index 165180
- Subject index 167182
- Back Cover Back Cover1184