Lecture Notes

The course notes were prepared jointly by Prof. Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik.

The book Tensor Categories based on these 2009 notes was published by the American Mathematical Society in 2015. A complete file of the book (PDF - 3.1MB) is on Prof. Etingof’s webpage. [Please note: This file cannot be posted on any website not belonging to the authors.]

Complete 2009 lecture notes in one file (PDF - 2.5MB)

SES # TOPICS LECTURE NOTES
1 Basics of monoidal categories (PDF)
2

Monoidal functors

MacLane’s strictness theorem

(PDF)
3

MacLane coherence theorem

Rigid monoidal categories

Invertible objects

Tensor and multitensor categories

(PDF)
4

Tensor product and tensor functors

Unit object

Grothendieck rings

Groupoids

Finite abelian categories

Fiber functors

Coalgebras

(PDF)
5 Bialgebras and Hopf algebras (PDF)
6

Quantum groups

Skew-primitive elements

Pointed tensor categories

Coradical filtration

Chevalley’s theorem and Chevalley property

(PDF)
7

Andruskeiwitsch-Schneider conjecture

Cartier-Kostant theorem

Quasi-bialgebras and quasi-Hopf algebras

(PDF)
8

Quantum traces

Pivotal categories and dimensions

Spherical categories

Multitensor cateogries

Multifusion rings

Frobenius-Perron theorem

(PDF)
9

Tensor categories

Deligne’s tensor product

Finite (multi)tensor categories

Categorical freeness

(PDF)
10

Distinguished invertible object

Integrals in quasi-Hopf algebras

Cartan matrix

Basics of Module categories

(PDF)
11

Exact module categories

Algebras in categories

Internal Hom

(PDF)
12

Main Theorem

Categories of module functors

Dual categories

(PDF)

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