Lie Groups and Algebraic Groups

Arkadij L. Onishchik, Ernest B. Vinberg (Autoren)

Buch | Softcover

XX, 330 Seiten

2011 | 1. Softcover reprint of the original 1st ed. 1990
Springer Berlin (Verlag)
978-3-642-74336-8 (ISBN)

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This book is based on the notes of the authors' seminar on algebraic and Lie groups held at the Department of Mechanics and Mathematics of Moscow University in 1967/68. Our guiding idea was to present in the most economic way the theory of semisimple Lie groups on the basis of the theory of algebraic groups. Our main sources were A. Borel's paper [34], C. ChevalIey's seminar [14], seminar "Sophus Lie" [15] and monographs by C. Chevalley [4], N. Jacobson [9] and J-P. Serre [16, 17]. In preparing this book we have completely rearranged these notes and added two new chapters: "Lie groups" and "Real semisimple Lie groups". Several traditional topics of Lie algebra theory, however, are left entirely disregarded, e.g. universal enveloping algebras, characters of linear representations and (co)homology of Lie algebras. A distinctive feature of this book is that almost all the material is presented as a sequence of problems, as it had been in the first draft of the seminar's notes. We believe that solving these problems may help the reader to feel the seminar's atmosphere and master the theory. Nevertheless, all the non-trivial ideas, and sometimes solutions, are contained in hints given at the end of each section. The proofs of certain theorems, which we consider more difficult, are given directly in the main text. The book also contains exercises, the majority of which are an essential complement to the main contents.

1. Lie Groups.-
1. Background.-
2. Tangent Algebra.-
3. Connectedness and Simple Connectedness.-
4. The Derived Algebra and the Radical.- 2. Algebraic Varieties.-
1. Affine Algebraic Varieties.-
2. Projective and Quasiprojective Varieties.-
3. Dimension and Analytic Properties of Algebraic Varieties.- 3. Algebraic Groups.-
1. Background.-
2. Commutative and Solvable Algebraic Groups.-
3. The Tangent Algebra.-
4. Compact Linear Groups.- 4. Complex Semisimple Lie Groups.-
1. Preliminaries.-
2. Root Systems.-
3. Existence and Uniqueness Theorems.-
4. Automorphisms.- 5. Real Semisimple Lie Groups.-
1. Real Forms of Complex Semisimple Lie Groups and Algebras.-
2. Compact Lie Groups and Reductive Algebraic Groups.-
3. Cartan Decomposition.-
4. Real Root Decomposition.- 6. Levi Decomposition.- 1°. Levi's Theorem.- 2°. Existence of a Lie Group with the Given Tangent Algebra.- 3°. Malcev's Theorem.- 4°. Algebraic Levi Decomposition.- Exercises.- Hints to Problems.- Reference Chapter.-
1. Useful Formulae.-
2. Tables.

Erscheint lt. Verlag	6.12.2011
Reihe/Serie	Springer Series in Soviet Mathematics
Übersetzer	Dimitry A. Leites
Zusatzinfo	XX, 330 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	539 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Themenwelt	Naturwissenschaften ► Physik / Astronomie
Schlagworte	Algebraic Geometry • algebraic group • Algebraische Gruppen • Darstellungstheorie • group theory • Lie-Algebra • Lie-Gruppe • Representation Theory • Tensorprodukt
ISBN-10	3-642-74336-6 / 3642743366
ISBN-13	978-3-642-74336-8 / 9783642743368
Zustand	Neuware