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Product Code:  SURV/205.S 
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Electronic ISBN:  9781470423490 
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Book DetailsMathematical Surveys and MonographsVolume: 205; 2015; 344 ppMSC: Primary 17; 18; 19; 20;
Is there a vector space whose dimension is the golden ratio? Of course not—the golden ratio is not an integer! But this can happen for generalizations of vector spaces—objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, lowdimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finitedimensional Hopf algebras may be derived from the theory of tensor categories.
Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.ReadershipGraduate students and research mathematicians interested in category theory and Hopf algebras.

Table of Contents

Chapters

Chapter 1. Abelian categories

Chapter 2. Monoidal categories

Chapter 3. $\mathbb {Z}_+$rings

Chapter 4. Tensor categories

Chapter 5. Representation categories of Hopf algebras

Chapter 6. Finite tensor categories

Chapter 7. Module categories

Chapter 8. Braided categories

Chapter 9. Fusion categories


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Is there a vector space whose dimension is the golden ratio? Of course not—the golden ratio is not an integer! But this can happen for generalizations of vector spaces—objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, lowdimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finitedimensional Hopf algebras may be derived from the theory of tensor categories.
Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.
Graduate students and research mathematicians interested in category theory and Hopf algebras.

Chapters

Chapter 1. Abelian categories

Chapter 2. Monoidal categories

Chapter 3. $\mathbb {Z}_+$rings

Chapter 4. Tensor categories

Chapter 5. Representation categories of Hopf algebras

Chapter 6. Finite tensor categories

Chapter 7. Module categories

Chapter 8. Braided categories

Chapter 9. Fusion categories