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This famous book was the first treatise on Lie groups in which a modern point of view was adopted systematically, namely, that a continuous group can be regarded as a global object. To develop this idea to its fullest extent, Chevalley incorporated a broad range of topics, such as the covering spaces of topological spaces, analytic manifolds, integration of complete systems of differential equations on a manifold, and the calculus of exterior differential forms.

The book opens with a short description of the classical groups: unitary groups, orthogonal groups, symplectic groups, etc. These special groups are then used to illustrate the general properties of Lie groups, which are considered later. The general notion of a Lie group is defined and correlated with the algebraic notion of a Lie algebra; the subgroups, factor groups, and homomorphisms of Lie groups are studied by making use of the Lie algebra. The last chapter is concerned with the theory of compact groups, culminating in Peter-Weyl's theorem on the existence of representations. Given a compact group, it is shown how one can construct algebraically the corresponding Lie group with complex parameters which appears in the form of a certain algebraic variety (associated algebraic group). This construction is intimately related to the proof of the generalization given by Tannaka of Pontrjagin's duality theorem for Abelian groups.

The continued importance of Lie groups in mathematics and theoretical physics make this an indispensable volume for researchers in both fields. ... Read more

Customer Reviews (1)

A landmark, but a bit dated.
This is really a landmark in textbooks on Lie groups, and once was "the" book to be read together with Pontryagin's "Topological Groups". Unfortunately, the writing style has become a bit "dated" (this was written in the early forties!). Of course, that doesn't mean Chevalley's book has become useless, but for the first reading, I would recommend Adams' book (Benjamin / Addison Wesley?) instead. ... Read more

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This volume is the first in a projected series devoted to the mathematical and philosophical works of the late Claude Chevalley. It covers the main contributions by the author to the theory of spinors. Since its appearance in 1954, "The Algebraic Theory of Spinors" has been a much sought after reference. It presents the whole story of one subject in a concise and especially clear manner. The reprint of the book is supplemented by a series of lectures on Clifford Algebras given by the author in Japan at about the same time. Also included is a postface by J.-P. Bourguignon describing the many uses of spinors in differential geometry developed by mathematical physicists from the 1970s to the present day. An insightful review of "Spinors" by J. Dieudonne is also made available to the reader in this new edition. ... Read more

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The third volume of the Collected Works of Claude Chevalley assembles his work on semi-simple algebraic groups contained, for the most part, in the notes of the famous "Séminaire Chevalley" held at the Ecole Normale Supérieure in Paris between 1956 and 1958 and written up by participants of the seminar namely, P. Cartier, A. Grothendieck, R. Lazard and J. L. Verdier.

These texts have been entirely reset in TeX for this edition, and edited and annotated by Pierre Cartier. Almost 50 years after the original writing, these texts still constitute a choice reference from which to enter and learn this part of the theory of algebraic groups.

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Chapters: Alexander Grothendieck, André Weil, Nicolas Bourbaki, Séminaire Nicolas Bourbaki, Serge Lang, Jean-Pierre Serre, Claude Chevalley, Laurent Schwartz, Henri Cartan, Jean Dieudonné, Jacques Tits, Samuel Eilenberg, Armand Borel, John Tate, Charles Ehresmann, Szolem Mandelbrojt, Pierre Samuel, René de Possel, Roger Godement, Jacques Dixmier, Pierre Cartier, Jean-Louis Koszul, Jean Delsarte, François Bruhat. Source: Wikipedia. Pages: 119. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is one of the most influential mathematicians of the twentieth century. He is known principally for his revolutionary advances in algebraic geometry, and also for major contributions to number theory, category theory and homological algebra, and his early achievements in functional analysis. He was awarded the Fields Medal in 1966. He was co-awarded the Crafoord Prize with Pierre Deligne in 1988, but Grothendieck declined this. He is noted for his mastery of abstract approaches to mathematics and his perfectionism in matters of formulation and presentation. In particular, he demonstrated the ability to derive concrete results using only very general methods. Relatively little of his work after 1960 was published by the conventional route of the learned journal, circulating initially in duplicated volumes of seminar notes; his influence was to a considerable extent personal, on French mathematics and the Zariski school at Harvard University. He retired in 1988 and within a few years became reclusive. Grothendieck's early mathematical work was done in functional analysis between 1949 and 1953 working on his doctoral thesis in Nancy, supervised by Jean Dieudonné and Laurent Schwartz. His key contributions include topological tensor products of to...More: http://booksllc.net/?id=2042 ... Read more

Product Description
This volume is the first in a projected series devoted to the mathematical and philosophical works of the late Claude Chevalley. It covers the main contributions by the author to the theory of spinors. Since its appearance in 1954, "The Algebraic Theory of Spinors" has been a much sought after reference. It presents the whole story of one subject in a concise and especially clear manner. The reprint of the book is supplemented by a series of lectures on Clifford Algebras given by the author in Japan at about the same time. Also included is a postface by J.-P. Bourguignon describing the many uses of spinors in differential geometry developed by mathematical physicists from the 1970s to the present day. An insightful review of "Spinors" by J. Dieudonne is also made available to the reader in this new edition. ... Read more

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This classic book, originally published in 1968, is based on notes of a year-long seminar the authors ran at Princeton University. The primary goal of the book was to give a rather complete presentation of algebraic aspects of global class field theory, and the authors accomplished this goal spectacularly: for more than 40 years since its first publication, the book has served as an ultimate source for many generations of mathematicians. In this revised edition, two mathematical additions complementing the exposition in the original text are made. The new edition also contains several new footnotes, additional references, and historical comments. ... Read more