18.769 | Spring 2009 | Graduate

Topics in Lie Theory: Tensor Categories

Readings

In order to prepare for class, students are required to read selections from the course notes. These readings can be found on the lecture notes page.

SES # TOPICS READINGS
1 Basics of monoidal categories Sections 1.1-1.3
2

Monoidal functors

MacLane’s strictness theorem

Sections 1.4-1.8
3

MacLane coherence theorem

Rigid monoidal categories

Invertible objects

Tensor and multitensor categories

Sections 1.9-1.12
4

Tensor product and tensor functors

Unit object

Grothendieck rings

Groupoids

Finite abelian categories

Fiber functors

Coalgebras

Sections 1.13-1.20
5 Bialgebras and Hopf algebras Sections 1.21-1.24
6

Quantum groups

Skew-primitive elements

Pointed tensor categories

Coradical filtration

Chevalley’s theorem and Chevalley property

Sections 1.25-1.31
7

Andruskeiwitsch-Schneider conjecture

Cartier-Kostant theorem

Quasi-bialgebras and quasi-Hopf algebras

Sections 1.32-1.36
8

Quantum traces

Pivotal categories and dimensions

Spherical categories

Multitensor cateogries

Multifusion rings

Frobenius-Perron theorem

Sections 1.37-1.44
9

Tensor categories

Deligne’s tensor product

Finite (multi)tensor categories

Categorical freeness

Sections 1.45-1.50
10

Distinguished invertible object

Integrals in quasi-Hopf algebras

Cartan matrix

Basics of Module categories

Sections 1.51-1.53 and 2.1-2.6
11

Exact module categories

Algebras in categories

Internal Hom

Sections 2.7-2.10
12

Main Theorem

Categories of module functors

Dual categories

Sections 2.11-2.14

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