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Representations of Semisimple Lie Algebras in the BGG Category $\mathscr {O}$

James E. Humphreys University of Massachusetts, Amherst, Amherst, MA
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Softcover ISBN: 978-1-4704-6326-7
Product Code: GSM/94.S
List Price: $69.00 MAA Member Price:$62.10
AMS Member Price: $55.20 Electronic ISBN: 978-1-4704-2120-5 Product Code: GSM/94.E List Price:$65.00
MAA Member Price: $58.50 AMS Member Price:$52.00
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List Price: $103.50 MAA Member Price:$93.15
AMS Member Price: $82.80 Click above image for expanded view Representations of Semisimple Lie Algebras in the BGG Category$\mathscr {O}$James E. Humphreys University of Massachusetts, Amherst, Amherst, MA Available Formats:  Softcover ISBN: 978-1-4704-6326-7 Product Code: GSM/94.S  List Price:$69.00 MAA Member Price: $62.10 AMS Member Price:$55.20
 Electronic ISBN: 978-1-4704-2120-5 Product Code: GSM/94.E
 List Price: $65.00 MAA Member Price:$58.50 AMS Member Price: $52.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$103.50 MAA Member Price: $93.15 AMS Member Price:$82.80
• Book Details

Volume: 942008; 289 pp
MSC: Primary 17; Secondary 20; 22;

This is the first textbook treatment of work leading to the landmark 1979 Kazhdan–Lusztig Conjecture on characters of simple highest weight modules for a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb {C}$. The setting is the module category $\mathscr {O}$ introduced by Bernstein–Gelfand–Gelfand, which includes all highest weight modules for $\mathfrak{g}$ such as Verma modules and finite dimensional simple modules. Analogues of this category have become influential in many areas of representation theory.

Part I can be used as a text for independent study or for a mid-level one semester graduate course; it includes exercises and examples. The main prerequisite is familiarity with the structure theory of $\mathfrak{g}$. Basic techniques in category $\mathscr {O}$ such as BGG Reciprocity and Jantzen's translation functors are developed, culminating in an overview of the proof of the Kazhdan–Lusztig Conjecture (due to Beilinson–Bernstein and Brylinski–Kashiwara). The full proof however is beyond the scope of this book, requiring deep geometric methods: $D$-modules and perverse sheaves on the flag variety. Part II introduces closely related topics important in current research: parabolic category $\mathscr {O}$, projective functors, tilting modules, twisting and completion functors, and Koszul duality theorem of Beilinson–Ginzburg–Soergel.

Graduate students and research mathematicians interested in Lie theory, and representation theory.

• Chapters
• Chapter 0. Review of semisimple Lie algebras
• Part 1. Highest weight modules
• Chapter 1. Category $\mathcal {O}$: Basics
• Chapter 2. Characters of finite dimensional modules
• Chapter 3. Category $\mathcal {O}$: Methods
• Chapter 4. Highest weight modules I
• Chapter 5. Highest weight modules II
• Chapter 6. Extensions and resolutions
• Chapter 7. Translation functors
• Chapter 8. Kazhdan-Lusztig theory
• Part 2. Further developments
• Chapter 9. Parabolic versions of category $\mathcal {O}$
• Chapter 10. Projective functors and principal series
• Chapter 11. Tilting modules
• Chapter 12. Twisting and completion functors
• Chapter 13. Complements

• Reviews

• One of the goals Humphreys had in mind was to provide a textbook suitable for graduate students. This has been achieved by keeping prerequisites to a minimum, by careful dealing with technical parts of the proofs, and by offering a large number of exercises.

Mathematical Reviews
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 942008; 289 pp
MSC: Primary 17; Secondary 20; 22;

This is the first textbook treatment of work leading to the landmark 1979 Kazhdan–Lusztig Conjecture on characters of simple highest weight modules for a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb {C}$. The setting is the module category $\mathscr {O}$ introduced by Bernstein–Gelfand–Gelfand, which includes all highest weight modules for $\mathfrak{g}$ such as Verma modules and finite dimensional simple modules. Analogues of this category have become influential in many areas of representation theory.

Part I can be used as a text for independent study or for a mid-level one semester graduate course; it includes exercises and examples. The main prerequisite is familiarity with the structure theory of $\mathfrak{g}$. Basic techniques in category $\mathscr {O}$ such as BGG Reciprocity and Jantzen's translation functors are developed, culminating in an overview of the proof of the Kazhdan–Lusztig Conjecture (due to Beilinson–Bernstein and Brylinski–Kashiwara). The full proof however is beyond the scope of this book, requiring deep geometric methods: $D$-modules and perverse sheaves on the flag variety. Part II introduces closely related topics important in current research: parabolic category $\mathscr {O}$, projective functors, tilting modules, twisting and completion functors, and Koszul duality theorem of Beilinson–Ginzburg–Soergel.

Graduate students and research mathematicians interested in Lie theory, and representation theory.

• Chapters
• Chapter 0. Review of semisimple Lie algebras
• Part 1. Highest weight modules
• Chapter 1. Category $\mathcal {O}$: Basics
• Chapter 2. Characters of finite dimensional modules
• Chapter 3. Category $\mathcal {O}$: Methods
• Chapter 4. Highest weight modules I
• Chapter 5. Highest weight modules II
• Chapter 6. Extensions and resolutions
• Chapter 7. Translation functors
• Chapter 8. Kazhdan-Lusztig theory
• Part 2. Further developments
• Chapter 9. Parabolic versions of category $\mathcal {O}$
• Chapter 10. Projective functors and principal series
• Chapter 11. Tilting modules
• Chapter 12. Twisting and completion functors
• Chapter 13. Complements
• One of the goals Humphreys had in mind was to provide a textbook suitable for graduate students. This has been achieved by keeping prerequisites to a minimum, by careful dealing with technical parts of the proofs, and by offering a large number of exercises.

Mathematical Reviews
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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