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OverviewThese notes, already well known in their original French edition, give the basic theory of semisimple Lie algebras over the complex numbers including the basic classification theorem. The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and representation theory. The theory is illustrated by using the example of sln; in particular, the representation theory of sl2 is completely worked out. The last chapter discusses the connection between Lie algebras and Lie groups, and is intended to guide the reader towards further study. Full Product DetailsAuthor: Jean-Pierre Serre , Glen JonesPublisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Imprint: Springer-Verlag Berlin and Heidelberg GmbH & Co. K Edition: 1st ed. 1987. Reprint 2000 Dimensions: Width: 15.50cm , Height: 0.60cm , Length: 23.50cm Weight: 0.454kg ISBN: 9783540678274ISBN 10: 3540678271 Pages: 75 Publication Date: 12 December 2000 Audience: College/higher education , Professional and scholarly , Postgraduate, Research & Scholarly , Professional & Vocational Format: Hardback Publisher's Status: Active Availability: Out of print, replaced by POD  We will order this item for you from a manufatured on demand supplier. Table of ContentsI Nilpotent Lie Algebras and Solvable Lie Algebras.- 1. Lower Central Series.- 2. Definition of Nilpotent Lie Algebras.- 3. An Example of a Nilpotent Algebra.- 4. Engel’s Theorems.- 5. Derived Series.- 6. Definition of Solvable Lie Algebras.- 7. Lie’s Theorem.- 8. Cartan’s Criterion.- II Semisimple Lie Algebras (General Theorems).- 1. Radical and Semisimpiicity.- 2. The Cartan-Killing Criterion.- 3. Decomposition of Semisimple Lie Algebras.- 4. Derivations of Semisimple Lie Algebras.- 5. Semisimple Elements and Nilpotent Elements.- 6. Complete Reducibility Theorem.- 7. Complex Simple Lie Algebras.- 8. The Passage from Real to Complex.- III Cartan Subalgebras.- 1. Definition of Cartan Subalgebras.- 2. Regular Elements: Rank.- 3. The Cartan Subalgebra Associated with a Regular Element.- 4. Conjugacy of Cartan Subalgebras.- 5. The Semisimple Case.- 6. Real Lie Algebras.- IV The Algebra SI2 and Its Representations.- 1. The Lie Algebra sl2.- 2. Modules, Weights, Primitive Elements.- 3. Structure of the Submodule Generated by a Primitive Element.- 4. The Modules Wm.- 5. Structure of the Finite-Dimensional g-Modules.- 6. Topological Properties of the Group SL2.- V Root Systems.- 1. Symmetries.- 2. Definition of Root Systems.- 3. First Examples.- 4. The Weyl Group.- 5. Invariant Quadratic Forms.- 6. Inverse Systems.- 7. Relative Position of Two Roots.- 8. Bases.- 9. Some Properties of Bases.- 10. Relations with the Weyl Group.- 11. The Cartan Matrix.- 12. The Coxeter Graph.- 13. Irreducible Root Systems.- 14. Classification of Connected Coxeter Graphs.- 15. Dynkin Diagrams.- 16. Construction of Irreducible Root Systems.- 17. Complex Root Systems.- VI Structure of Semisimple Lie Algebras.- 1. Decomposition of g.- 2. Proof of Theorem 2.- 3. Borei Subalgebras.- 4. WeylBases.- 5. Existence and Uniqueness Theorems.- 6. Chevalley’s Normalization.- Appendix. Construction of Semisimple Lie Algebras by Generators and Relations.- VII Linear Representations of Semisimple Lie Algebras.- 1. Weights.- 2. Primitive Elements.- 3. Irreducible Modules with a Highest Weight.- 4. Finite-Dimensional Modules.- 5. An Application to the Weyl Group.- 6. Example: sl n+1.- 7. Characters.- 8. H. Weyl’s formula.- VIII Complex Groups and Compact Groups.- 1. Cartan Subgroups.- 2. Characters.- 3. Relations with Representations.- 4. Berel Subgroups.- 5. Construction of Irreducible Representations from Boret Subgroups.- 6. Relations with Algebraic Groups.- 7. Relations with Compact Groups.ReviewsFrom the reviews of the French edition: <p>.,. the book is intended for those who have an acquaintance with the basic parts of the theory, namely, with those general theorems on Lie algebras which do not depend on the notion of Cartan subalgebra. The author begins with a summary of these general theorems and then discusses in detail the structure and representation theory of complex semisimple Lie algebras. One recognizes here a skillful ordering of the material, many simplifications of classical arguments and a new theorem describing fundamental relations between canonical generators of semisimple Lie algebras. The classical theory being thus introduced in such modern form, the reader can quickly reach the essence of the theory through the present book. (Mathematical Reviews) From the reviews of the French edition: ...the book is intended for those who have an acquaintance with the basic parts of the theory, namely, with those general theorems on Lie algebras which do not depend on the notion of Cartan subalgebra. The author begins with a summary of these general theorems and then discusses in detail the structure and representation theory of complex semisimple Lie algebras. One recognizes here a skillful ordering of the material, many simplifications of classical arguments and a new theorem describing fundamental relations between canonical generators of semisimple Lie algebras. The classical theory being thus introduced in such modern form, the reader can quickly reach the essence of the theory through the present book. (Mathematical Reviews) From the reviews of the French edition: ...the book is intended for those who have an acquaintance with the basic parts of the theory, namely, with those general theorems on Lie algebras which do not depend on the notion of Cartan subalgebra. The author begins with a summary of these general theorems and then discusses in detail the structure and representation theory of complex semisimple Lie algebras. One recognizes here a skillful ordering of the material, many simplifications of classical arguments and a new theorem describing fundamental relations between canonical generators of semisimple Lie algebras. The classical theory being thus introduced in such modern form, the reader can quickly reach the essence of the theory through the present book. (Mathematical Reviews) Author InformationTab Content 6Author Website:Countries AvailableAll regions |
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