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Books > Science & Mathematics > Mathematics > Topology > Algebraic topology
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Essays in the History of Lie Groups and Algebraic Groups (Hardcover)
Loot Price: R3,396
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Essays in the History of Lie Groups and Algebraic Groups (Hardcover)
Series: History of Mathematics
Expected to ship within 12 - 17 working days
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Algebraic groups and Lie groups are important in most major areas
of mathematics, occuring in diverse roles such as the symmetries of
differential equations and as central figures in the Langlands
program for number theory. In this book, Professor Borel looks at
the development of the theory of Lie groups and algebraic groups,
highlighting the evolution from the almost purely local theory at
the start to the global theory that we know today. As the starting
point of this passage from local to global, the author takes Lie's
theory of local analytic transformation groups and Lie algebras. He
then follows the globalization of the process in its two most
important frameworks: (transcendental) differential geometry and
algebraic geometry. 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 $SL 2M$, the contributions H. Weyl to
representation and invariant theory for 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 starts with
Chapter V describing the development of the theory of linear
algebraic groups in the 19th century. Many of the main
contributions here are due to E. Study, E. Cartan, and above all,
to L. Maurer. After being abandoned for nearly 50 years, the theory
was revived by Chevalley and Kolchin and then further developed by
many others. This is the focus of Chapter VI. The book concludes
with two chapters on various aspects of the works of Chevalley on
Lie groups and algebraic groups and Kolchin on algebraic groups and
the Galois theory of differential fields. The author 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
appreciation of the underlying mathematics. His lifelong
involvement and his historical research in the subject give him a
special appreciation of the story of its development.
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