主题目录

    • University:  Massachusetts Institute of Technology

      Instructors:  Prof. Pavel Etingof

      Course Number:  18.769

      Level:  Graduate

      Course Description

      This course will give a detailed introduction to the theory of tensor categories and review some of its connections to other subjects (with a focus on representation-theoretic applications). In particular, we will discuss categorifications of such notions from ring theory as: module, morphism of modules, Morita equivalence of rings, commutative ring, the center of a ring, the centralizer of a subring, the double centralizer property, graded ring, etc.

  • 教学大纲(Syllabus)

    • Prerequisites

      Before beginning this course, you are expected to know basic category theory, basic algebra, and fundamentals of group representations. Hopf algebras will often appear in the course, but no previous knowledge of them is expected.

  • 其他教学资源(Other Resources)

    • 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 #TOPICSREADINGS
      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
  • 作业

    Exercises can be found in the lecture notes.

    SES #TOPICSASSIGNMENTS
    1 Basics of monoidal categories Exercise 1.2.5
    2

    Monoidal functors

    MacLane's strictness theorem

    Exercises 1.4.4, 1.7.5, 1.8.8
    3

    MacLane coherence theorem

    Rigid monoidal categories

    Invertible objects

    Tensor and multitensor categories

    Exercises 1.10.5, 1.10.6, 1.10.8, 1.10.15, 1.10.16
    4

    Tensor product and tensor functors

    Unit object

    Grothendieck rings

    Groupoids

    Finite abelian categories

    Fiber functors

    Coalgebras

    Exercises 1.15.3, 1.15.6, 1.15.10, 1.17.1, 1.17.3, 1.18.3, 1.19.3, 1.20.3, 1.20.4
    5 Bialgebras and Hopf algebras Exercises 1.21.4, 1.21.5, 1.21.6, 1.22.3, 1.22.12, 1.22.14, 1.22.16, 1.22.17, 1.22.18, 1.24.3, 1.24.4, 1.24.8, 1.24.10, 1.24.11
    6

    Quantum groups

    Skew-primitive elements

    Pointed tensor categories

    Coradical filtration

    Chevalley's theorem and Chevalley property

    Exercises 1.25.3, 1.29.2, 1.31.7
    7

    Andruskeiwitsch-Schneider conjecture

    Cartier-Kostant theorem

    Quasi-bialgebras and quasi-Hopf algebras

    Exercises 1.34.9, 1.35.3, 1.36.3
    8

    Quantum traces

    Pivotal categories and dimensions

    Spherical categories

    Multitensor cateogries

    Multifusion rings

    Frobenius-Perron theorem

    Exercises 1.37.2, 1.27.4, 1.38.3, 1.39.3, 1.42.7
    9

    Tensor categories

    Deligne's tensor product

    Finite (multi)tensor categories

    Categorical freeness

    Exercises 1.45.14, 1.49.2
    10

    Distinguished invertible object

    Integrals in quasi-Hopf algebras

    Cartan matrix

    Basics of Module categories

    Exercises 1.52.9, 2.1.4, 2.1.7, 2.5.3, 2.5.8, 2.6.2, 2.6.7
    11

    Exact module categories

    Algebras in categories

    Internal Hom

    Exercises 2.8.2, 2.8.4, 2.8.6, 2.8.8, 2.9.7, 2.9.8, 2.9.9, 2.9.11, 2.9.13, 2.9.13, 2.9.15, 2.9.16, 2.9.17, 2.9.20, 2.9.23, 2.9.26, 2.10.3, 2.10.9
    12

    Main Theorem

    Categories of module functors

    Dual categories

    Exercises 2.11.1, 2.11.4, 2.11.5, 2.12.4, 2.13.1