Humphreys, James E. Introduction to Lie Algebras and Representation Theory

# Introduction to Lie Algebras and Representation Theory

## Humphreys, James E.

Valutazione media 4,38
( su 16 valutazioni fornite da GoodReads )

This book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations. A good knowledge of linear algebra (including eigenvalues, bilinear forms, euclidean spaces, and tensor products of vector spaces) is presupposed, as well as some acquaintance with the methods of abstract algebra. The first four chapters might well be read by a bright undergraduate; however, the remaining three chapters are admittedly a little more demanding. Besides being useful in many parts of mathematics and physics, the theory of semisimple Lie algebras is inherently attractive, combining as it does a certain amount of depth and a satisfying degree of completeness in its basic results. Since Jacobson's book appeared a decade ago, improvements have been made even in the classical parts of the theory. I have tried to incor- porate some of them here and to provide easier access to the subject for non-specialists. For the specialist, the following features should be noted: (I) The Jordan-Chevalley decomposition of linear transformations is emphasized, with "toral" subalgebras replacing the more traditional Cartan subalgebras in the semisimple case. (2) The conjugacy theorem for Cartan subalgebras is proved (following D. J. Winter and G. D. Mostow) by elementary Lie algebra methods, avoiding the use of algebraic geometry.

Le informazioni nella sezione "Riassunto" possono far riferimento a edizioni diverse di questo titolo.

Contenuti:

I. Basic Concepts.- 1. Definitions and first examples.- 1.1 The notion of Lie algebra.- 1.2 Linear Lie algebras.- 1.3 Lie algebras of derivations.- 1.4 Abstract Lie algebras.- 2. Ideals and homomorphisms.- 2.1 Ideals.- 2.2 Homomorphisms and representations.- 2.3 Automorphisms.- 3. Solvable and nilpotent Lie algebras.- 3.1 Solvability.- 3.2 Nilpotency.- 3.3 Proof of Engel’s Theorem.- II. Semisimple Lie Algebras.- 4. Theorems of Lie and Cartan.- 4.1 Lie’s Theorem.- 4.2 Jordan-Chevalley decomposition.- 4.3 Cartan’s Criterion.- 5. Killing form.- 5.1 Criterion for semisimplicity.- 5.2 Simple ideals of L.- 5.3 Inner derivations.- 5.4 Abstract Jordan decomposition.- 6. Complete reducibility of representations.- 6.1 Modules.- 6.2 Casimir element of a representation.- 6.3 Weyl’s Theorem.- 6.4 Preservation of Jordan decomposition.- 7. Representations of sl (2, F).- 7.1 Weights and maximal vectors.- 7.2 Classification of irreducible modules.- 8. Root space decomposition.- 8.1 Maximal toral subalgebras and roots.- 8.2 Centralizer of H.- 8.3 Orthogonality properties.- 8.4 Integrality properties.- 8.5 Rationality properties Summary.- III. Root Systems.- 9. Axiomatics.- 9.1 Reflections in a euclidean space.- 9.2 Root systems.- 9.3 Examples.- 9.4 Pairs of roots.- 10. Simple roots and Weyl group.- 10.1 Bases and Weyl chambers.- 10.2 Lemmas on simple roots.- 10.3 The Weyl group.- 10.4 Irreducible root systems.- 11. Classification.- 11.1 Cartan matrix of ?.- 11.2 Coxeter graphs and Dynkin diagrams.- 11.3 Irreducible components.- 11.4 Classification theorem.- 12. Construction of root systems and automorphisms.- 12.1 Construction of types A-G.- 12.2 Automorphisms of ?.- 13. Abstract theory of weights.- 13.1 Weights.- 13.2 Dominant weights.- 13.3 The weight ?.- 13.4 Saturated sets of weights.- IV. Isomorphism and Conjugacy Theorems.- 14. Isomorphism theorem.- 14.1 Reduction to the simple case.- 14.2 Isomorphism theorem.- 14.3 Automorphisms.- 15. Cartan subalgebras.- 15.1 Decomposition of L relative to ad x.- 15.2 Engel subalgebras.- 15.3 Cartan subalgebras.- 15.4 Functorial properties.- 16. Conjugacy theorems.- 16.1 The group g (L).- 16.2 Conjugacy of CSA’s (solvable case).- 16.3 Borel subalgebras.- 16.4 Conjugacy of Borel subalgebras.- 16.5 Automorphism groups.- V. Existence Theorem.- 17. Universal enveloping algebras.- 17.1 Tensor and symmetric algebras.- 17.2 Construction of U(L).- 17.3 PBW Theorem and consequences.- 17.4 Proof of PBW Theorem.- 17.5 Free Lie algebras.- 17. Generators and relations.- 17.1 Relations satisfied by L.- 17.2 Consequences of (S1)-(S3).- 17.3 Serre’s Theorem.- 17.4 Application: Existence and uniqueness theorems.- 18. The simple algebras.- 18.1 Criterion for semisimplicity.- 18.2 The classical algebras.- 18.3 The algebra G2.- VI. Representation Theory.- 20. Weights and maximal vectors.- 20.1 Weight spaces.- 20.2 Standard cyclic modules.- 20.3 Existence and uniqueness theorems.- 21. Finite dimensional modules.- 21.1 Necessary condition for finite dimension.- 21.2 Sufficient condition for finite dimension.- 21.3 Weight strings and weight diagrams.- 21.4 Generators and relations for V(?).- 22. Multiplicity formula.- 22.1 A universal Casimir element.- 22.2 Traces on weight spaces.- 22.3 Freudenthal’s formula.- 22.4 Examples.- 22.5 Formal characters.- 23. Characters.- 23.1 Invariant polynomial functions.- 23.2 Standard cyclic modules and characters.- 23.3 Harish-Chandra’s Theorem.- 24. Formulas of Weyl, Kostant, and Steinberg.- 24.1 Some functions on H*.- 24.2 Kostant’s multiplicity formula.- 24.3 Weyl’s formulas.- 24.4 Steinberg’s formula.- VII. Chevalley Algebras and Groups.- 25. Chevalley basis of L.- 25.1 Pairs of roots.- 25.2 Existence of a Chevalley basis.- 25.3 Uniqueness questions.- 25.4 Reduction modulo a prime.- 25.5 Construction of Chevalley groups (adjoint type).- 26. Kostant’s Theorem.- 26.1 A combinatorial lemma.- 26.2 Special case: sl (2, F).- 26.3 Lemmas on commutation.- 26.4 Proof of Kostant’s Theorem.- 27. Admissible lattices.- 27.1 Existence of admissible lattices.- 27.2 Stabilizer of an admissible lattice.- 27.3 Variation of admissible lattice.- 27.4 Passage to an arbitrary field.- 27.5 Survey of related results.- References.- Afterword (1994).- Index of Terminology.- Index of Symbols.

Le informazioni nella sezione "Su questo libro" possono far riferimento a edizioni diverse di questo titolo.

### I migliori risultati di ricerca su AbeBooks

Edizione Internazionale

## 1.Introduction to Lie Algebras and Representation Theory

Editore: Springer Verlag, China (2008)
ISBN 10: 0387900535 ISBN 13: 9780387900537
Nuovi Soft cover Quantità: 1
Edizione Internazionale
Da
Bonny. Lee
Valutazione libreria

Descrizione libro Springer Verlag, China, 2008. Soft cover. Condizione libro: Brand New. 12mo - over 6¾ - 7¾" tall. International edition Brand New SOFTCOVER standard deliver. Codice libro della libreria 000914

Compra nuovo
EUR 21,37
Convertire valuta
Spese di spedizione: EUR 27,45
Da: Canada a: U.S.A.
Destinazione, tempi e costi
Edizione Internazionale

## 2.Introduction to Lie Algebras and Representation Theory

Editore: Springer Verlag, China (2008)
ISBN 10: 0387900535 ISBN 13: 9780387900537
Nuovi Soft cover Quantità: 1
Edizione Internazionale
Da
Bonny. Lee
Valutazione libreria

Descrizione libro Springer Verlag, China, 2008. Soft cover. Condizione libro: Brand New. 12mo - over 6¾ - 7¾" tall. International edition Brand New SOFCOVER standard delivery. Codice libro della libreria 001088

Compra nuovo
EUR 21,37
Convertire valuta
Spese di spedizione: EUR 27,45
Da: Canada a: U.S.A.
Destinazione, tempi e costi

## 3.Introduction to Lie Algebras and Representation Theory: v. 9

Editore: Springer-Verlag New York Inc. (1994)
ISBN 10: 0387900535 ISBN 13: 9780387900537
Nuovi Quantità: > 20
Print on Demand
Da
PBShop
(Secaucus, NJ, U.S.A.)
Valutazione libreria

Descrizione libro Springer-Verlag New York Inc., 1994. HRD. Condizione libro: New. New Book. Shipped from US within 10 to 14 business days. THIS BOOK IS PRINTED ON DEMAND. Established seller since 2000. Codice libro della libreria I1-9780387900537

Compra nuovo
EUR 59,12
Convertire valuta
Spese di spedizione: EUR 3,65
In U.S.A.
Destinazione, tempi e costi

## 4.Introduction to Lie Algebras and Representation Theory

Editore: Springer (1973)
ISBN 10: 0387900535 ISBN 13: 9780387900537
Nuovi Rilegato Quantità: 1
Da
Herb Tandree Philosophy Books
(Stroud, GLOS, Regno Unito)
Valutazione libreria

Descrizione libro Springer, 1973. Hardback. Condizione libro: NEW. 9780387900537 Hardback, This listing is a new book, a title currently in-print which we order directly and immediately from the publisher. Codice libro della libreria HTANDREE0275079

Compra nuovo
EUR 54,29
Convertire valuta
Spese di spedizione: EUR 8,97
Da: Regno Unito a: U.S.A.
Destinazione, tempi e costi

## 5.Introduction to Lie Algebras and Representation Theory (Graduate Texts in Mathematics) (v. 9)

Editore: Springer (1994)
ISBN 10: 0387900535 ISBN 13: 9780387900537
Nuovi Rilegato Quantità: 1
Da
Book Deals
(Lewiston, NY, U.S.A.)
Valutazione libreria

Descrizione libro Springer, 1994. Condizione libro: New. Brand New, Unread Copy in Perfect Condition. A+ Customer Service!. Codice libro della libreria ABE_book_new_0387900535

Compra nuovo
EUR 65,14
Convertire valuta
Spese di spedizione: GRATIS
In U.S.A.
Destinazione, tempi e costi

## 6.Introduction to Lie Algebras and Representation Theory

Editore: Springer (2016)
ISBN 10: 0387900535 ISBN 13: 9780387900537
Nuovi Paperback Quantità: 1
Print on Demand
Da
Ria Christie Collections
(Uxbridge, Regno Unito)
Valutazione libreria

Descrizione libro Springer, 2016. Paperback. Condizione libro: New. PRINT ON DEMAND Book; New; Publication Year 2016; Not Signed; Fast Shipping from the UK. No. book. Codice libro della libreria ria9780387900537_lsuk

Compra nuovo
EUR 65,13
Convertire valuta
Spese di spedizione: EUR 3,23
Da: Regno Unito a: U.S.A.
Destinazione, tempi e costi

## 7.Introduction to Lie Algebras and Representation Theory

Editore: Springer 1997-04-01 (1997)
ISBN 10: 0387900535 ISBN 13: 9780387900537
Nuovi Quantità: 5
Da
Chiron Media
(Wallingford, Regno Unito)
Valutazione libreria

Descrizione libro Springer 1997-04-01, 1997. Condizione libro: New. Brand new book, sourced directly from publisher. Dispatch time is 24-48 hours from our warehouse. Book will be sent in robust, secure packaging to ensure it reaches you securely. Codice libro della libreria NU-ING-00699587

Compra nuovo
EUR 66,62
Convertire valuta
Spese di spedizione: EUR 3,35
Da: Regno Unito a: U.S.A.
Destinazione, tempi e costi

## 8.Introduction to Lie Algebras and Representation Theory (Graduate Texts in Mathematics) (v. 9)

Editore: Springer (1973)
ISBN 10: 0387900535 ISBN 13: 9780387900537
Nuovi Rilegato Quantità: 1
Print on Demand
Da
Ergodebooks
(RICHMOND, TX, U.S.A.)
Valutazione libreria

Descrizione libro Springer, 1973. Hardcover. Condizione libro: New. This item is printed on demand. Codice libro della libreria DADAX0387900535

Compra nuovo
EUR 66,67
Convertire valuta
Spese di spedizione: EUR 3,65
In U.S.A.
Destinazione, tempi e costi

## 9.Introduction to Lie Algebras and Representation Theory (Graduate Texts in Mathematics) (v. 9)

Editore: Springer (1994)
ISBN 10: 0387900535 ISBN 13: 9780387900537
Nuovi Rilegato Quantità: 1
Da
Irish Booksellers
(Rumford, ME, U.S.A.)
Valutazione libreria

Descrizione libro Springer, 1994. Hardcover. Condizione libro: New. book. Codice libro della libreria 0387900535

Compra nuovo
EUR 72,13
Convertire valuta
Spese di spedizione: GRATIS
In U.S.A.
Destinazione, tempi e costi

## 10.Introduction to Lie Algebras and Representation Theory: v. 9 (Hardback)

Editore: Springer-Verlag New York Inc., United States (1994)
ISBN 10: 0387900535 ISBN 13: 9780387900537
Nuovi Rilegato Quantità: 10
Print on Demand
Da
The Book Depository
(London, Regno Unito)
Valutazione libreria

Descrizione libro Springer-Verlag New York Inc., United States, 1994. Hardback. Condizione libro: New. 1st ed. 1972. Corr. 7th printing 1994. 232 x 160 mm. Language: English . Brand New Book ***** Print on Demand *****.This book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations. A good knowledge of linear algebra (including eigenvalues, bilinear forms, euclidean spaces, and tensor products of vector spaces) is presupposed, as well as some acquaintance with the methods of abstract algebra. The first four chapters might well be read by a bright undergraduate; however, the remaining three chapters are admittedly a little more demanding. Besides being useful in many parts of mathematics and physics, the theory of semisimple Lie algebras is inherently attractive, combining as it does a certain amount of depth and a satisfying degree of completeness in its basic results. Since Jacobson s book appeared a decade ago, improvements have been made even in the classical parts of the theory. I have tried to incor- porate some of them here and to provide easier access to the subject for non-specialists. For the specialist, the following features should be noted: (I) The Jordan-Chevalley decomposition of linear transformations is emphasized, with toral subalgebras replacing the more traditional Cartan subalgebras in the semisimple case. (2) The conjugacy theorem for Cartan subalgebras is proved (following D. J. Winter and G. D. Mostow) by elementary Lie algebra methods, avoiding the use of algebraic geometry. Codice libro della libreria APC9780387900537

Compra nuovo
EUR 79,22
Convertire valuta
Spese di spedizione: GRATIS
Da: Regno Unito a: U.S.A.
Destinazione, tempi e costi