**Mathematical Surveys and Monographs**

Volume: 168;
2010;
411 pp;
Hardcover

MSC: Primary 16; 05; 20; 17; 14; 81;
Secondary 58; 82

Print ISBN: 978-0-8218-5262-0

Product Code: SURV/168

List Price: $109.00

Individual Member Price: $87.20

**Electronic ISBN: 978-1-4704-1395-8
Product Code: SURV/168.E**

List Price: $109.00

Individual Member Price: $87.20

#### Supplemental Materials

# Algebras, Rings and Modules: Lie Algebras and Hopf Algebras

Share this page
*Michiel Hazewinkel; Nadiya Gubareni; V. V. Kirichenko*

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.

#### Readership

Research mathematicians interested in Hopf algebras and Lie algebras.

#### Table of Contents

# Table of Contents

## Algebras, Rings and Modules: Lie Algebras and Hopf Algebras

- Contents v6 free
- Preface ix10 free
- Chapter 1. Lie algebras and Dynkin diagrams 114 free
- Chapter 2. Coalgebras: motivation, definitions, and examples 99112
- Chapter 3. Bialgebras and Hopf algebras. Motivation, definitions, and examples 131144
- Chapter 4. The Hopf algebra of symmetric functions 175188
- Chapter 5. The representations of the symmetric groups from the Hopf algebra point of view 217230
- Chapter 6. The Hopf algebra of noncommutative symmetric functions and the Hopf algebra of quasisymmetric functions 231244
- Chapter 7. The Hopf algebra of permutations 263276
- Chapter 8. Hopf algebras: Applications in and interrelations with other parts of mathematics and physics 277290
- Index 407420 free