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Algebras, Rings and Modules: Lie Algebras and Hopf Algebras

Nadiya Gubareni Technical University of Czȩstochowa, Czȩstochowa, Poland
V. V. Kirichenko Kiev National Taras Shevchenko University, Kiev, Ukraine
Available Formats:
Hardcover ISBN: 978-0-8218-5262-0
Product Code: SURV/168
List Price: $116.00 MAA Member Price:$104.40
AMS Member Price: $92.80 Electronic ISBN: 978-1-4704-1395-8 Product Code: SURV/168.E List Price:$109.00
MAA Member Price: $98.10 AMS Member Price:$87.20
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List Price: $174.00 MAA Member Price:$156.60
AMS Member Price: $139.20 Click above image for expanded view Algebras, Rings and Modules: Lie Algebras and Hopf Algebras Nadiya Gubareni Technical University of Czȩstochowa, Czȩstochowa, Poland V. V. Kirichenko Kiev National Taras Shevchenko University, Kiev, Ukraine Available Formats:  Hardcover ISBN: 978-0-8218-5262-0 Product Code: SURV/168  List Price:$116.00 MAA Member Price: $104.40 AMS Member Price:$92.80
 Electronic ISBN: 978-1-4704-1395-8 Product Code: SURV/168.E
 List Price: $109.00 MAA Member Price:$98.10 AMS Member Price: $87.20 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:$174.00 MAA Member Price: $156.60 AMS Member Price:$139.20
• Book Details

Mathematical Surveys and Monographs
Volume: 1682010; 411 pp
MSC: Primary 16; 05; 20; 17; 14; 81;

The main goal of this book is to present an introduction to and applications of the theory of Hopf algebras. The authors also discuss some important aspects of the theory of Lie algebras.

The first chapter can be viewed as a primer on Lie algebras, with the main goal to explain and prove the Gabriel–Bernstein–Gelfand–Ponomarev theorem on the correspondence between the representations of Lie algebras and quivers; this material has not previously appeared in book form.

The next two chapters are also “primers” on coalgebras and Hopf algebras, respectively; they aim specifically to give sufficient background on these topics for use in the main part of the book. Chapters 4–7 are devoted to four of the most beautiful Hopf algebras currently known: the Hopf algebra of symmetric functions, the Hopf algebra of representations of the symmetric groups (although these two are isomorphic, they are very different in the aspects they bring to the forefront), the Hopf algebras of the nonsymmetric and quasisymmetric functions (these two are dual and both generalize the previous two), and the Hopf algebra of permutations. The last chapter is a survey of applications of Hopf algebras in many varied parts of mathematics and physics.

Unique features of the book include a new way to introduce Hopf algebras and coalgebras, an extensive discussion of the many universal properties of the functor of the Witt vectors, a thorough discussion of duality aspects of all the Hopf algebras mentioned, emphasis on the combinatorial aspects of Hopf algebras, and a survey of applications already mentioned. The book also contains an extensive (more than 700 entries) bibliography.

Research mathematicians interested in Hopf algebras and Lie algebras.

• Chapters
• 1. Lie algebras and Dynkin diagrams
• 2. Coalgebras: Motivation, definitions, and examples
• 3. Bialgebras and Hopf algebras. Motivation, definitions, and examples
• 4. The Hopf algebra of symmetric functions
• 5. The representations of the symmetric groups from the Hopf algebra point of view
• 6. The Hopf algebra of noncommutative symmetric functions and the Hopf algebra of quasisymmetric functions
• 7. The Hopf algebra of permutations
• 8. Hopf algebras: Applications in and interrelations with other parts of mathematics and physics

• 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: 1682010; 411 pp
MSC: Primary 16; 05; 20; 17; 14; 81;

The main goal of this book is to present an introduction to and applications of the theory of Hopf algebras. The authors also discuss some important aspects of the theory of Lie algebras.

The first chapter can be viewed as a primer on Lie algebras, with the main goal to explain and prove the Gabriel–Bernstein–Gelfand–Ponomarev theorem on the correspondence between the representations of Lie algebras and quivers; this material has not previously appeared in book form.

The next two chapters are also “primers” on coalgebras and Hopf algebras, respectively; they aim specifically to give sufficient background on these topics for use in the main part of the book. Chapters 4–7 are devoted to four of the most beautiful Hopf algebras currently known: the Hopf algebra of symmetric functions, the Hopf algebra of representations of the symmetric groups (although these two are isomorphic, they are very different in the aspects they bring to the forefront), the Hopf algebras of the nonsymmetric and quasisymmetric functions (these two are dual and both generalize the previous two), and the Hopf algebra of permutations. The last chapter is a survey of applications of Hopf algebras in many varied parts of mathematics and physics.

Unique features of the book include a new way to introduce Hopf algebras and coalgebras, an extensive discussion of the many universal properties of the functor of the Witt vectors, a thorough discussion of duality aspects of all the Hopf algebras mentioned, emphasis on the combinatorial aspects of Hopf algebras, and a survey of applications already mentioned. The book also contains an extensive (more than 700 entries) bibliography.

Research mathematicians interested in Hopf algebras and Lie algebras.

• Chapters
• 1. Lie algebras and Dynkin diagrams
• 2. Coalgebras: Motivation, definitions, and examples
• 3. Bialgebras and Hopf algebras. Motivation, definitions, and examples
• 4. The Hopf algebra of symmetric functions
• 5. The representations of the symmetric groups from the Hopf algebra point of view
• 6. The Hopf algebra of noncommutative symmetric functions and the Hopf algebra of quasisymmetric functions
• 7. The Hopf algebra of permutations
• 8. Hopf algebras: Applications in and interrelations with other parts of mathematics and physics
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
Please select which format for which you are requesting permissions.