Quantum Groups and Their Representations (Theoretical and Mathematical Physics) by Anatoli Klimyk (PDF)

Description

This book provides a treatment of the theory of quantum groups (quantized universal enveloping algebras and quantized algebras of functions) and q-deformed algebras (q-oscillator algebras), their representations and corepresentations, and noncommutative differential calculus. The theory of the simplest and most important quantum groups and their representations is presented in detail. A number of topics and results from the more advanced general theory are developed and discussed. Many applications in mathematical and theoretical physics are indicated. The book starts as an introduction for the beginner and continues at a textbook level for graduate students in physics and in mathematics. It may serve as a reference for more advanced readers.

User’s Reviews

Editorial Reviews: Review From the reviews”Klimyk and Schmüdgen are kind to their readers. Proofs are given in full, and there are helpful explanations of the basic concepts … the book has the virtue of comprehensivness in its chose range of topics. It is easy to dip into and use as a reference book.” (A. Sudbery, Bulletin of the London Mathematical Society, 2000) From the Back Cover This book start with an introduction to quantum groups for the beginner and continues as a textbook for graduate students in physics and in mathematics. It can also be used as a reference by more advanced readers. The authors cover a large but well-chosen variety of subjects from the theory of quantum groups (quantized universal enveloping algebras, quantized algebras of functions) and q-deformed algebras (q-oscillator algebras), their representations and corepresentations, and noncommutative differential calculus. The book is written with potential applications in physics and mathematics in mind. The basic quantum groups and quantum algebras and their representations are given in detail and accompanied by explicit formulas. A number of topics and results from the more advanced general theory are developed and discussed.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a very good and solid introduction to, and reference book on, quantum groups (or quantum algebras, as they are also known). It is one of the more clear volumes on the subject, and presents information in a fairly straightforward way.As with many introductions to quantum groups, it can take a while to understand what is happening unless one already has some understanding of the field, however, the writing is not obtuse.One of the notable aspects of it (for me) was that it clearly presents the differences between the Drinfeld-Jimbo version of the quantum group (defined over the ring of power series in an indeterminate) and Jimbo’s version of the quantum group (defined over the complex field). The Drinfeld-Jimbo quantum group admits a universal R-matrix while the Jimbo version does not – for the latter, one can define solutions of the matrix Yang-Baxter equation from any integrable finite dimensional representation but this must be done carefully, and this volume presents this problem clearly.I recommend this volume for graduate students working on quantum groups; I referred to it many times in my Ph.D studies, and also to people working with quantum groups.

⭐A wonderful book

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