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.