Basic Theory of Algebraic Groups and Lie Algebras
(Graduate Texts in Mathematics 75)
by: G. P. Hochschild
Product Details
* Hardcover: 267 pages
* Publisher: Springer; 1 edition (June 30, 1981)
* Language: English
* ISBN-10: 0387905413
* ISBN-13: 978-0387905419
Preface
The theory of algebraic groups results from the interaction of various basic
techniques from field theory, multilinear algebra, commutative ring theory,
algebraic geometry and general algebraic representation theory of groups and
Lie algebras. It is thus an ideally suitable framework for exhibiting basic
algebra in action. To do that is the principal concern of this text. Accordingly,
its emphasis is on developing the major general mathematical tools used for
gaining control over algebraic groups, rather than on securing the final
definitive results, such as the classification of the simple groups and their
irreducible representations. In the same spirit, this exposition has been made
entirely self-contained; no detailed knowledge beyond the usual standard
material of the first one or two years of graduate study in algebra is presupposed.
The chapter headings should be sufficient indication of the content and
organisation of this book. Each chapter begins with a brief announcement of
its results and ends with a few notes ranging from supplementary results,
amplifications of proofs, examples and counter-examples through exercises
to references. The references are intended to be merely suggestions for
supplementary reading or indications of original sources, especially in cases
where these might not be the expected ones.
Algebraic group theory has reached a state of maturity and perfection
where it may no longer be necessary to re-iterate an account ofits genesis. Of
the material to be presented here, including much of the basic support, the
major portion is due to Claude Chevalley. Although Chevalley's decisive
classification results, contained in [6], have not been included here, a
glimpse of their main ingredients can be had from Chapters XVII and XIII.
The subject of Chapter XIII is Armand Borel's fundamental theory of
possible to recreate the combinatorial features of the Cartan-Weyl theory of
semisimple Lie algebras, dealt with in Chapter XVII, in terms ofsubgroups of
semisimple algebraic groups. In particular, this has freed the theory from the
classical restriction to base fields of characteristic 0.
I was encouraged to write this exposition chiefly by the appearance of
James Humphreys's Linear Algebraic Groups, where the required algebraic
geometry has been cut down to a manageable size. In fact, the algebraic geometric
developments given here have resulted from Humphreys's
treatment simply by adding proofs ofthe underlying facts from commutative
algebra. Moreover, much ofthe general structure theory in arbitrary characteristic
has been adapted from Borel's lecture notes and Humphreys's
book
ifile.it
or
megaupload.com
(Graduate Texts in Mathematics 75)
by: G. P. Hochschild
Product Details
* Hardcover: 267 pages
* Publisher: Springer; 1 edition (June 30, 1981)
* Language: English
* ISBN-10: 0387905413
* ISBN-13: 978-0387905419
Preface
The theory of algebraic groups results from the interaction of various basic
techniques from field theory, multilinear algebra, commutative ring theory,
algebraic geometry and general algebraic representation theory of groups and
Lie algebras. It is thus an ideally suitable framework for exhibiting basic
algebra in action. To do that is the principal concern of this text. Accordingly,
its emphasis is on developing the major general mathematical tools used for
gaining control over algebraic groups, rather than on securing the final
definitive results, such as the classification of the simple groups and their
irreducible representations. In the same spirit, this exposition has been made
entirely self-contained; no detailed knowledge beyond the usual standard
material of the first one or two years of graduate study in algebra is presupposed.
The chapter headings should be sufficient indication of the content and
organisation of this book. Each chapter begins with a brief announcement of
its results and ends with a few notes ranging from supplementary results,
amplifications of proofs, examples and counter-examples through exercises
to references. The references are intended to be merely suggestions for
supplementary reading or indications of original sources, especially in cases
where these might not be the expected ones.
Algebraic group theory has reached a state of maturity and perfection
where it may no longer be necessary to re-iterate an account ofits genesis. Of
the material to be presented here, including much of the basic support, the
major portion is due to Claude Chevalley. Although Chevalley's decisive
classification results, contained in [6], have not been included here, a
glimpse of their main ingredients can be had from Chapters XVII and XIII.
The subject of Chapter XIII is Armand Borel's fundamental theory of
possible to recreate the combinatorial features of the Cartan-Weyl theory of
semisimple Lie algebras, dealt with in Chapter XVII, in terms ofsubgroups of
semisimple algebraic groups. In particular, this has freed the theory from the
classical restriction to base fields of characteristic 0.
I was encouraged to write this exposition chiefly by the appearance of
James Humphreys's Linear Algebraic Groups, where the required algebraic
geometry has been cut down to a manageable size. In fact, the algebraic geometric
developments given here have resulted from Humphreys's
treatment simply by adding proofs ofthe underlying facts from commutative
algebra. Moreover, much ofthe general structure theory in arbitrary characteristic
has been adapted from Borel's lecture notes and Humphreys's
book
ifile.it
or
megaupload.com
